############################################################################# ## #A groupid.grp GAP group library Frank Celler #A & Hans-Georg Esser ## #A @(#)$Id: groupid.grp,v 3.6.1.2 1994/08/18 16:01:56 fceller Rel $ ## #Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany ## ## The information in the file is based on the Neubueser Catalogue for ## groups of order less or equal than 100, excluding groups of order 64 and ## 96, the Catalogue of M.Hall and J.K.Senior for groups of order 64, and ## the R.Laue Catalogue of groups of order 96. ## #H $Log: groupid.grp,v $ #H Revision 3.6.1.2 1994/08/18 16:01:56 fceller #H added '3primes' for small groups #H #H Revision 3.6.1.1 1994/08/18 13:42:49 fceller #H added pqr groups #H #H Revision 3.6 1994/06/03 11:58:47 fceller #H added abelian 2- and 3-groups #H #H Revision 3.5 1994/06/02 09:15:31 fceller #H added 2 and 3 groups #H #H Revision 3.4 1994/06/02 09:10:22 fceller #H initial GAP 3.4 revision ## if not IsBound(StandardPresentation) then StandardPresentation := 0; fi; ############################################################################# ## #F NrElementsOfOrder( , ) . . number of elements of of order ## NrElementsOfOrder := function( G, n ) return Number( Elements(G), x -> Order(G,x) = n ); end; ############################################################################# ## #F EncodedStandardPresentation( ,

) . . . . . encode pcp presentation ## ## This is the inverse of 'ThGGroup' and 'TGGroup'. ## EncodedStandardPresentation := function ( G, p ) local c, n, rhs, i, k, enc, mul, l, rank; # get number of generators c := Cgs(G); n := Length(c); # construct right hand sides of the presentation rhs := []; for i in [ 1 .. n ] do rhs[i] := []; rhs[i][i] := ExponentsAgWord(c[i]^p); for k in [ 1 .. i-1 ] do rhs[i][k] := ExponentsAgWord(Comm(c[i],c[k])); od; od; # encode presentation enc := 0; mul := 1; for i in [ 1 .. n ] do for k in [1..i-1] do for l in [k..i-1] do enc := enc + rhs[l][k][i]*mul; mul := mul*p; od; od; od; # get rank if IsBound(G.rank) then rank := G.rank; else rank := LogInt( Index(G,FrattiniSubgroup(G)), p ); fi; return [ rank, n, Length(PCentralSeries(G,p))-1, enc ]; end; ############################################################################# ## #F GroupIdOps . . . . . . . . . . . . . . . . . . . . . . operations record ## GroupIdOps := OperationsRecord( "GroupIdOps" ); ############################################################################# ## #V GroupIdOps.PGroups . . . . . . . . . . . . . . p-group catalogue number ## GroupIdOps.PGroups := []; GroupIdOps.PGroups[ 2] := [1]; GroupIdOps.PGroups[ 4] := [2,1]; GroupIdOps.PGroups[ 8] := [5,2,1,3,4]; GroupIdOps.PGroups[16] := [14,10,5,2,1,11,12,13,3,4,6,7,8,9]; GroupIdOps.PGroups[32] := [51,45,36,21,16,3,1,46,47,48,22,23,37,25,26,24,38, 2,4,5,12,17,39,40,41,42,9,10,14,13,11,15,27,34, 35,28,29,30,31,32,33,49, 50,43,44,6,7,8,18,19,20]; GroupIdOps.PGroups[64] := [267,260,246,192,183,83,50,55,26,2,1,261,262,263, 193,194,247,196,197,195,248,56,84,87,103,184,198,58,59,57,85,86,112,115, 126,185,17,3,27,29,44,51,250,251,252,253,95,96,107,106,101,110,97,108, 118,119,120,124,21,20,22,6,7,16,15,31,45,202,211,212,203,204,205,207, 208,209,206,213,214,210,60,65,61,67,71,66,72,70,62,63,69,68,64,88,104, 89,105,113,116,117,127,114,264,265,266,199,201,200,249,254,255,256,99, 98,100,109,123,121,122,102,111,125,90,92,93,91,94,23,24,25,4,5,30,186, 187,188,189,38,39,47,48,40,49,73,76,75,74,80,79,77,78,81,82,226,230,239, 227,231,229,228,235,238,232,234,240,236,233,237,215,216,218,217,224, 225,219,221,220,223,222,18,19,28,242,241,243,244,245,174,173,181,179, 175,167,168,147,146,148,176,180,169,128,131,129,132,140,141,155,142, 157,156,143,158,161,162,164,165,130,133,144,145,159,160,163,166,177, 178,182,150,149,151,171,170,172,12,13,14,8,9,10,11,257,258,259,190,191, 46,42,41,43,32,33,34,35,36,37,153,152,154,138,139,134,136,135,137,52, 53,54]; GroupIdOps.PGroups[ 3] := [1]; GroupIdOps.PGroups[ 9] := [2,1]; GroupIdOps.PGroups[27] := [5,2,1,3,4]; GroupIdOps.PGroups[81] := [15,11,5,2,1,12,13,3,4,14,6,7,9,8,10]; ############################################################################# ## #V GroupIdOps.3primes . . . . . . . . . . . . pqr group id for small groups ## GroupIdOps.3primes := []; GroupIdOps.3primes[18] := [,,["M",2,3],["D",2,3,3],["K",2,3]]; GroupIdOps.3primes[30] := [,["D",2,5,3],["D",2,3,5],["G",2,3,5,1]]; GroupIdOps.3primes[42] := [,["D",2,7,3],["D",3,7,2],["D",2,3,7],["H",2,3,7], ["G",2,3,7,1]]; GroupIdOps.3primes[50] := [,,["M",2,5],["D",2,5,5],["K",2,5]]; GroupIdOps.3primes[66] := [,["D",2,11,3],["D",2,3,11],["G",2,3,11,1]]; GroupIdOps.3primes[70] := [,["D",2,7,5],["D",2,5,7],["G",2,5,7,1]]; GroupIdOps.3primes[75] := [,,["N",3,5]]; GroupIdOps.3primes[78] := [,["D",2,13,3],["D",3,13,2],["D",2,3,13], ["H",2,3,13],["G",2,3,13,1]]; GroupIdOps.3primes[98] := [,,["M",2,7],["D",2,7,7],["K",2,7]]; ############################################################################# ## #F GroupIdOps.AbelianPGroups . . . . . . . abelian p-group catalogue number ## GroupIdOps.AbelianPGroups := []; GroupIdOps.AbelianPGroups[128] := [ [[128],1],[[8,16],42],[[4,32],128],[[2,64],159],[[2,8,8],179], [[4,4,8],456],[[2,4,16],837],[[2,2,32],988],[[2,4,4,4],997], [[2,2,4,8],1601],[[2,2,2,16],2136],[[2,2,2,4,4],2150], [[2,2,2,2,8],2301],[[2,2,2,2,2,4],2319],[[2,2,2,2,2,2,2],2328] ]; GroupIdOps.AbelianPGroups[256] := [ [[256],1],[[16,16],39],[[8,32],316],[[4,64],497],[[2,128],537], [[4,8,8],826],[[2,8,16],4384],[[4,4,16],5525],[[2,4,32],6534], [[2,2,64],6723],[[4,4,4,4],6732],[[2,2,8,8],10298],[[2,4,4,8],13313], [[2,2,4,16],26308],[[2,2,2,32],26959],[[2,2,4,4,4],26973], [[2,2,2,4,8],53038],[[2,2,2,2,16],55608],[[2,2,2,2,4,4],55626], [[2,2,2,2,2,8],56059],[[2,2,2,2,2,2,4],56082],[[2,2,2,2,2,2,2,2],56092] ]; GroupIdOps.AbelianPGroups[243] := [ [[243],1],[[9,27],10],[[3,81],23],[[3,9,9],31],[[3,3,27],48], [[3,3,3,9],61],[[3,3,3,3,3],67] ]; GroupIdOps.AbelianPGroups[729] := [ [[729],1],[[27,27],2],[[9,81],58],[[3,243],93],[[9,9,9],102], [[3,9,27],238],[[3,3,81],391],[[3,3,9,9],415],[[3,3,3,27],475], [[3,3,3,3,9],497],[[3,3,3,3,3,3],504] ]; ############################################################################# ## #F GroupIdOps.Nr4Squares( ) . different squares of elements of order 4 ## GroupIdOps.Nr4Squares := function( G ) local tmp; # returns a list of all squares of elements of order 4. tmp := List( Collected( List( Filtered( Elements(G), x -> Order(G,x)=4 ), x -> x^2 ) ), z -> z[2] ); Sort(tmp); return tmp; end; ############################################################################# ## #F GroupIdOps.Nr8Squares( ) . different squares of elements of order 8 ## GroupIdOps.Nr8Squares := function( G ) local tmp; # returns a list of all squares of elements of order 16. tmp := List( Collected( List( Filtered( Elements(G), x -> Order(G,x)=8 ), x -> x^2 ) ), z -> z[2] ); Sort(tmp); return tmp; end; ############################################################################# ## #F GroupIdOps.NrClasses( , , ) . . . . number of conjugacy classes ## GroupIdOps.NrClasses := function( G, l, n ) local c; # number of conjugacy classes of length with element order c := ConjugacyClasses(G); c := Filtered( c, x -> Size(x) = l ); return Number( c, x -> Order(G,Representative(x)) = n ); end; ############################################################################# ## #F GroupIdOps.Stable( , ,

) . number of stable conjugacy classes ## GroupIdOps.Stable := function( G, o, p ) local c1; # get conjugacy classes of elements of order c1 := Filtered( Elements(G), x -> Order(G,x) = o ); c1 := Orbits( G, c1 ); # check how many are mapped to themself under x -> x ^

return Number( c1, x -> x[1]^p in x ); end; ############################################################################# ## #F GroupIdOps.Stable2( ) . . . . . . number of stable conjugacy classes ## GroupIdOps.Stable2 := function( G ) local c1, c2; c1 := Filtered( Elements(G), x -> Order(G,x) = 2 ); c1 := Orbits( G, c1 ); c2 := Filtered( Elements(G), x -> Order(G,x) = 4 ); c2 := List( Orbits( G, c2 ), x -> x[1]^2 ); c2 := List( c2, x -> First( c1, t -> x in t ) ); c2 := List( Collected(List(c2,x->Position(c1,x))), x -> x[2] ); Sort(c2); return c2; end; ############################################################################# ## #F GroupIdOps.Group16( ) . . . . . . . . . . . . . . groups of order 16 ## GroupIdOps.Group16 := function( G ) local x, nr, inv, name; # count the orders x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); # abelian groups if IsAbelian(G) then # 2 x 2 x 2 x 2 if x[4] = 15 then nr := 1; inv := [ 2, 2, 2, 2 ]; name := [ "2^4" ]; # 2 x 2 x 4 elif x[4] = 7 then nr := 2; inv := [ 2, 2, 4 ]; name := [ "2^2x4" ]; # 16 elif x[4] = 1 then nr := 5; inv := [ 16 ]; name := [ "16" ]; # 4 x 4 (warning: 16.3 and 16.4 as enumerated in GAP) elif x[4] = 3 and x[6] = 12 then nr := 4; inv := [ 4, 4 ]; name := [ "4x4" ]; # 2 x 8 (in the catalogue 16.3 and 16.4 are swapped) else # x[4] = 3 and x[6] = 4 the nr := 3; inv := [ 2, 8 ]; name := [ "2x8" ]; fi; # and return return rec( catalogue := [ 16, nr ], size := 16, names := name ); # non-abelian groups else name := []; # D8x2 if x[4] = 11 then nr := 6; name := [ "D8x2" ]; # no special name elif x[4] = 3 and x[6] = 4 then nr := 11; # D16 elif x[4] = 9 then nr := 12; name := [ "D16" ]; # quasi-dihedreal elif x[4] = 5 then nr := 13; name := [ "QD16" ]; # Q8x2 or (2x4).2 elif x[4] = 3 and x[6] =12 then if GroupIdOps.Nr4Squares(G) = [12] then nr := 7; name := [ "Q8x2" ]; else nr := 10; name := [ "(2x4).2" ]; fi; # D8Y4 or something else elif x[4] = 7 then if GroupIdOps.Nr4Squares(G) = [8] then nr := 8; name := [ "D8Y4" ]; else nr := 9; fi; # Q16 else nr := 14; name := [ "Q16" ]; fi; # and return return rec( catalogue := [ 16, nr ], size := 16, names := name ); fi; end; ############################################################################# ## #F GroupIdOps.Group18( ) . . . . . . . . . . . . . . groups of order 18 ## GroupIdOps.Group18 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList18, x ); # and return return rec( catalogue := [ 18, nr ], names := GroupIdOps.GroupName18[nr], size := 18 ); end; GroupIdOps.GroupName18 := [ ["6x3"], ["18"], ["D18"], ["S3x3"], ["3^2:2"] ]; GroupIdOps.GroupList18 := [ [ 1, 1, 2, 1, 3, 8, 6, 8 ], [ 1, 1, 2, 1, 3, 2, 6, 2, 9, 6, 18, 6 ], [ 1, 1, 2, 9, 3, 2, 9, 6 ], [ 1, 1, 2, 3, 3, 8, 6, 6 ], [ 1, 1, 2, 9, 3, 8 ] ]; ############################################################################# ## #F GroupIdOps.Group24( ) . . . . . . . . . . . . . . groups of order 24 ## GroupIdOps.Group24 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList24, x ); # and return return rec( catalogue := [ 24, nr ], names := GroupIdOps.GroupName24[nr], size := 24 ); end; GroupIdOps.GroupName24 := [ ["2^3x3"], ["12x2"], ["24"], ["D8x3"], ["Q8x3"], ["S3x2^2"], ["S3x4"], ["2x6.2"], ["12.2"], ["A4x2"], [], ["D24"], ["Q8+S3"], ["SL(2,3)"], ["S4"] ]; GroupIdOps.GroupList24 := [ [ 1, 1, 2, 7, 3, 2, 6, 14 ], [ 1, 1, 2, 3, 3, 2, 4, 4, 6, 6, 12, 8 ], [ 1, 1, 2, 1, 3, 2, 4, 2, 6, 2, 8, 4, 12, 4, 24, 8 ], [ 1, 1, 2, 5, 3, 2, 4, 2, 6, 10, 12, 4 ], [ 1, 1, 2, 1, 3, 2, 4, 6, 6, 2, 12, 12 ], [ 1, 1, 2, 15, 3, 2, 6, 6 ], [ 1, 1, 2, 7, 3, 2, 4, 8, 6, 2, 12, 4 ], [ 1, 1, 2, 3, 3, 2, 4, 12, 6, 6 ], [ 1, 1, 2, 1, 3, 2, 4, 2, 6, 2, 8, 12, 12, 4 ], [ 1, 1, 2, 7, 3, 8, 6, 8 ], [ 1, 1, 2, 9, 3, 2, 4, 6, 6, 6 ], [ 1, 1, 2, 13, 3, 2, 4, 2, 6, 2, 12, 4 ], [ 1, 1, 2, 1, 3, 2, 4, 14, 6, 2, 12, 4 ], [ 1, 1, 2, 1, 3, 8, 4, 6, 6, 8 ], [ 1, 1, 2, 9, 3, 8, 4, 6 ] ]; ############################################################################# ## #F GroupIdOps.Group30( ) . . . . . . . . . . . . . . groups of order 30 ## GroupIdOps.Group30 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList30, x ); # and return return rec( catalogue := [ 30, nr ], names := GroupIdOps.GroupName30[nr], size := 30 ); end; GroupIdOps.GroupName30 := [ [ "30" ], [ "D10x3" ], [ "S3x5" ], [ "D30" ] ]; GroupIdOps.GroupList30 := [ [ 1, 1, 2, 1, 3, 2, 5, 4, 6, 2, 10, 4, 15, 8, 30, 8 ], [ 1, 1, 2, 5, 3, 2, 5, 4, 6, 10, 15, 8 ], [ 1, 1, 2, 3, 3, 2, 5, 4, 10, 12, 15, 8 ], [ 1, 1, 2, 15, 3, 2, 5, 4, 15, 8 ] ]; ############################################################################# ## #F GroupIdOps.Group32( ) . . . . . . . . . . . . . . groups of order 32 ## GroupIdOps.Group32 := function( G ) local c, o, id4, nr; # compute the conjugacy classes # List(ConjugacyClasses(G),x->[Size(x),Order(G,Representative(x))]); c := Orbits( G, Elements(G) ); c := List( c, x -> [ Length(x), Order(G,x[1]) ] ); Sort(c); # and compute the element orders o := Collected( List( Elements(G), x -> Order(G,x) ) ); # and number of different squares of elements of order 4 id4 := GroupIdOps.Nr4Squares(G); # find these triples in our list nr := Position( GroupIdOps.GroupList32, Flat([o,id4,c]) ); # there are two problem pairs 13/20 and 19/21 if nr = 13 then if GroupIdOps.Nr8Squares(G) = [4,4,4,4] then nr := 20; fi; elif nr = 19 then if GroupIdOps.Nr8Squares(G) = [8,8] then nr := 21; fi; fi; # and return return rec( catalogue := [ 32, nr ], names := GroupIdOps.GroupName32[nr], size := 32 ); end; GroupIdOps.GroupName32 := [ ["2^5"], ["2^3x4"], ["2^2x8"], ["4^2x2"], ["16x2"], ["8x4"], ["32"], ["D8x2^2"], ["Q8x2^2"], ["D8Y4x2"], [], ["2x(2x4).2"], [], ["D8x4"], ["Q8x4"], [], ["D8Y8"], [], [], [], ["8+Q8"], [], ["D16x2"], ["QD16x2"], ["Q16x2"], ["D16Y4"], [], [], [], [], [], [], [], [], ["Q8+Q8"], [], [], [], [], [], [], ["D8YD8"], ["D8YQ8"], [], [], [], [], [], ["D32"], ["QD32"], ["Q32"]]; GroupIdOps.GroupList32 := [ [1,1,2,31,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1, 2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2], [1,1,2,15,4,16,16,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1, 2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4, 1,4,1,4,1,4], [1,1,2,7,4,8,8,16,8,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1, 8,1,8,1,8,1,8], [1,1,2,7,4,24,8,8,8,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1, 4,1,4,1,4,1,4], [1,1,2,3,4,4,8,8,16,16,4,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,8,1,8,1, 8,1,8,1,8,1,8,1,8,1,8,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16, 1,16,1,16,1,16,1,16,1,16,1,16,1,16], [1,1,2,3,4,12,8,16,4,4,4,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8, 1,8,1,8,1,8,1,8,1,8], [1,1,2,1,4,2,8,4,16,8,32,16,2,1,1,1,2,1,4,1,4,1,8,1,8,1,8,1,8,1,16, 1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,32,1,32,1,32,1,32,1,32,1,32,1, 32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32], [1,1,2,23,4,8,8,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,4,2,4,2,4,2,4], [1,1,2,7,4,24,24,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4], [1,1,2,15,4,16,16,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,2,2,2,2, 2,2,2,2,4,2,4,2,4,2,4,2,4,2,4], [1,1,2,15,4,16,8,8,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,2,2,2,2,2,2,2, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4], [1,1,2,7,4,24,8,16,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4], [1,1,2,7,4,8,8,16,8,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4, 2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8], [1,1,2,11,4,20,4,4,12,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,2,2, 2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4], [1,1,2,3,4,28,4,12,12,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4], [1,1,2,7,4,24,8,8,8,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4], [1,1,2,7,4,8,8,16,8,1,1,1,2,1,4,1,4,1,8,1,8,1,8,1,8,2,2,2,2,2,2,2,4, 2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8], [1,1,2,7,4,24,8,8,8,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4], [1,1,2,3,4,12,8,16,4,4,4,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2, 4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8], [1,1,2,7,4,8,8,16,8,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4, 2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8], [1,1,2,3,4,12,8,16,4,4,4,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2, 4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8], [1,1,2,3,4,4,8,8,16,16,4,1,1,1,2,1,4,1,4,1,8,1,8,1,8,1,8,2,2,2,4,2, 8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16], [1,1,2,19,4,4,8,8,4,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,4,2,4,2, 4,2,4,2], [1,1,2,11,4,12,8,8,12,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,4,2,4, 2,4,4,4,4], [1,1,2,3,4,20,8,8,20,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,4,4,4, 4,4,4,4,4], [1,1,2,11,4,12,8,8,12,1,1,1,2,1,4,1,4,2,2,2,4,2,8,2,8,2,8,2,8,4,2,4, 2,4,4,4,4], [1,1,2,11,4,12,8,8,4,8,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,4,2, 4,2,4,4,4,4], [1,1,2,3,4,20,8,8,8,12,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,4,4, 4,4,4,4,4,4], [1,1,2,3,4,20,8,8,4,16,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,4,4, 4,4,4,4,4,4], [1,1,2,3,4,20,8,8,4,8,8,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,4,4, 4,4,4,4,4,4], [1,1,2,7,4,16,8,8,4,4,8,1,1,1,2,1,4,1,4,2,2,2,4,2,4,2,4,2,4,2,4,4,2, 4,4,4,8,4,8], [1,1,2,3,4,4,8,24,4,1,1,1,2,1,4,1,4,2,2,2,4,2,8,2,8,2,8,2,8,4,8,4,8, 4,8,4,8], [1,1,2,19,4,12,4,4,4,1,1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4, 4,4,4,4,4], [1,1,2,19,4,12,4,4,4,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4, 2,4,2,4,2], [1,1,2,3,4,28,4,4,20,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4, 4,4,4,4,4], [1,1,2,15,4,16,8,8,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2, 4,4,4,4], [1,1,2,7,4,24,8,16,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,4,4,4, 4,4,4,4], [1,1,2,11,4,20,4,4,12,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4, 4,4,4,4,4], [1,1,2,11,4,20,4,4,12,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4, 2,4,4,4,4], [1,1,2,3,4,28,4,12,12,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4, 4,4,4,4,4], [1,1,2,7,4,24,8,8,8,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,4, 4,4,4,4], [1,1,2,19,4,12,12,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2, 4,2,4,2,4,2,4,2,4], [1,1,2,11,4,20,20,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4], [1,1,2,15,4,8,8,8,8,1,1,1,2,2,2,2,4,2,4,4,2,4,2,4,2,4,4,4,8,4,8], [1,1,2,7,4,16,8,8,16,1,1,1,2,2,2,2,4,2,4,4,2,4,4,4,4,4,4,4,8,4,8], [1,1,2,11,4,20,4,4,4,4,4,1,1,1,2,2,2,2,2,2,2,4,2,4,4,4,4,4,4,4,4,4,4], [1,1,2,11,4,4,8,16,4,1,1,1,2,2,2,2,4,2,4,4,2,4,2,4,8,4,8,4,8,4,8], [1,1,2,3,4,12,8,16,12,1,1,1,2,2,2,2,4,2,4,4,4,4,4,4,8,4,8,4,8,4,8], [1,1,2,17,4,2,8,4,16,8,2,1,1,1,2,2,4,2,8,2,8,2,16,2,16,2,16,2,16,8, 2,8,2], [1,1,2,9,4,10,8,4,16,8,10,1,1,1,2,2,4,2,8,2,8,2,16,2,16,2,16,2,16,8, 2,8,4], [1,1,2,1,4,18,8,4,16,8,18,1,1,1,2,2,4,2,8,2,8,2,16,2,16,2,16,2,16,8, 4,8,4]]; ############################################################################# ## #F GroupIdOps.Group36( ) . . . . . . . . . . . . . . groups of order 36 ## GroupIdOps.Group36 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList36, x ); # and return return rec( catalogue := [ 36, nr ], names := GroupIdOps.GroupName36[nr], size := 36 ); end; GroupIdOps.GroupName36 := [ ["6x6"], ["18x2"], ["12x3"], ["36"], ["6xS3"], ["3x6.2"], ["A4x3"], ["(2^2x3).3"], ["2x3^2:2"], ["(3^2x2).2"], ["D36"], ["18.2"], ["S3^2"], ["3^2:4"] ]; GroupIdOps.GroupList36 := [ [ 1, 1, 2, 3, 3, 8, 6, 24 ], [ 1, 1, 2, 3, 3, 2, 6, 6, 9, 6, 18, 18 ], # 36.3 in catalogue [ 1, 1, 2, 1, 3, 8, 4, 2, 6, 8, 12, 16 ], # 36.2 in catalogue [ 1, 1, 2, 1, 3, 2, 4, 2, 6, 2, 9, 6, 12, 4, 18, 6, 36, 12 ], [ 1, 1, 2, 7, 3, 8, 6, 20 ], [ 1, 1, 2, 1, 3, 8, 4, 6, 6, 8, 12, 12 ], [ 1, 1, 2, 3, 3, 26, 6, 6 ], [ 1, 1, 2, 3, 3, 2, 6, 6, 9, 24 ], [ 1, 1, 2, 19, 3, 8, 6, 8 ], [ 1, 1, 2, 1, 3, 8, 4, 18, 6, 8 ], [ 1, 1, 2, 19, 3, 2, 6, 2, 9, 6, 18, 6 ], [ 1, 1, 2, 1, 3, 2, 4, 18, 6, 2, 9, 6, 18, 6 ], [ 1, 1, 2, 15, 3, 8, 6, 12 ], [ 1, 1, 2, 9, 3, 8, 4, 18 ] ]; ############################################################################# ## #F GroupIdOps.Group40( ) . . . . . . . . . . . . . . groups of order 40 ## GroupIdOps.Group40 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList40, x ); # and return return rec( catalogue := [ 40, nr ], names := GroupIdOps.GroupName40[nr], size := 40 ); end; GroupIdOps.GroupName40 := [ ["2^3x5"], ["2x20"], ["40"], ["D8x5"], ["Q8x5"], ["2xD20"], ["4xD10"], ["2x10.2"], ["20.2"], ["2x5:4"], ["10.4"], [], ["D40"], ["Q8+D10"] ]; GroupIdOps.GroupList40 := [ [ 1, 1, 2, 7, 5, 4, 10, 28 ], [ 1, 1, 2, 3, 4, 4, 5, 4, 10, 12, 20, 16 ], [ 1, 1, 2, 1, 4, 2, 5, 4, 8, 4, 10, 4, 20, 8, 40, 16 ], [ 1, 1, 2, 5, 4, 2, 5, 4, 10, 20, 20, 8 ], [ 1, 1, 2, 1, 4, 6, 5, 4, 10, 4, 20, 24 ], [ 1, 1, 2, 23, 5, 4, 10, 12 ], [ 1, 1, 2, 11, 4, 12, 5, 4, 10, 4, 20, 8 ], [ 1, 1, 2, 3, 4, 20, 5, 4, 10, 12 ], [ 1, 1, 2, 1, 4, 2, 5, 4, 8, 20, 10, 4, 20, 8 ], [ 1, 1, 2, 11, 4, 20, 5, 4, 10, 4 ], [ 1, 1, 2, 1, 4, 10, 5, 4, 8, 20, 10, 4 ], [ 1, 1, 2, 13, 4, 10, 5, 4, 10, 12 ], [ 1, 1, 2, 21, 4, 2, 5, 4, 10, 4, 20, 8 ], [ 1, 1, 2, 1, 4, 22, 5, 4, 10, 4, 20, 8 ] ]; ############################################################################# ## #F GroupIdOps.Group42( ) . . . . . . . . . . . . . . groups of order 42 ## GroupIdOps.Group42 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList42, x ); # and return return rec( catalogue := [ 42, nr ], names := GroupIdOps.GroupName42[nr], size := 42 ); end; GroupIdOps.GroupName42 := [ [ "42" ], [ "D14x3" ], [ "7:3x2" ], [ "S3x7" ], [ "7:6" ], [ "D42" ] ]; GroupIdOps.GroupList42 := [ [ 1, 1, 2, 1, 3, 2, 6, 2, 7, 6, 14, 6, 21, 12, 42, 12 ], [ 1, 1, 2, 7, 3, 2, 6, 14, 7, 6, 21, 12 ], [ 1, 1, 2, 1, 3, 14, 6, 14, 7, 6, 14, 6 ], [ 1, 1, 2, 3, 3, 2, 7, 6, 14, 18, 21, 12 ], [ 1, 1, 2, 7, 3, 14, 6, 14, 7, 6 ], [ 1, 1, 2, 21, 3, 2, 7, 6, 21, 12 ] ]; ############################################################################# ## #F GroupIdOps.Group48( ) . . . . . . . . . . . . . . groups of order 48 ## GroupIdOps.Group48 := function( G ) local z1, z2, z3, nr, tmp; # look up element orders in group list z1 := Collected( List( Elements(G), x -> Order(G,x) ) ); # abelian or nor abelian if IsAbelian(G) then z2 := 1; else z2 := 0; fi; # length of conjugacy classes # List(ConjugacyClasses(G),x->[Size(x),Order(G,Representative(x))]) z3 := List( Orbits( G, Elements(G) ), x -> [Length(x),Order(G,x[1])] ); z3 := Collected(z3); Sort(z3); # get possible catalogue number nr := Position( GroupIdOps.GroupList48, Flat([z1,z2,z3]) ); # for 7/10 check 4 squares, 48.7 = [12], 48.10 = [4,8] if nr = 7 then if GroupIdOps.Nr4Squares(G) = [4,8] then nr := 10; fi; fi; # for 26/29/30 check 4 squares, 48.26=[28], 48.29=[12,16], 48.30=[4,24] if nr = 26 then tmp := GroupIdOps.Nr4Squares(G); if tmp = [12,16] then nr := 29; elif tmp = [4,24] then nr := 30; fi; fi; # and return return rec( catalogue := [ 48, nr ], names := GroupIdOps.GroupName48[nr], size := 48 ); end; GroupIdOps.GroupName48 := [ ["2^4x3"], ["2^2x12"], ["24x2"], ["4^2x3"], ["48"], ["6xD8"], ["6xQ8"], ["3xD8Y4"], [ ], ["3x(2x4).2"], [], ["3xD16"], ["3xQD16"], ["3xQ16"], ["2^3xS3"], ["2x4xS3"], ["2^2x6.2"], ["4x6.2"], ["8xS3"], ["2x12.2"], ["24.2"], ["2^2xA4"], ["4xA4"], [], ["2xD24"], ["2xQ8+S3"], [], [], [], [], [], [], [], ["SL(2,3)x2"], [], ["2xS4"], ["4+S4"], ["D8xS3"], ["Q8xS3"], ["(6.2)YD8"], ["(6.2)YQ8"], ["D48"], [], [], [], [], [], [], ["GL(2,3)"], [], ["A4+A4"], [] ]; GroupIdOps.GroupList48 := [ [1,1,2,15,3,2,6,30,1,1,1,1,1,2,15,1,3,2,1,6,30], [1,1,2,7,3,2,4,8,6,14,12,16,1,1,1,1,1,2,7,1,3,2,1,4,8,1,6,14,1,12,16], [1,1,2,3,3,2,4,4,6,6,8,8,12,8,24,16,1,1,1,1,1,2,3,1,3,2,1,4,4,1,6,6,1, 8,8,1,12,8,1,24,16], [1,1,2,3,3,2,4,12,6,6,12,24,1,1,1,1,1,2,3,1,3,2,1,4,12,1,6,6,1,12,24], [1,1,2,1,3,2,4,2,6,2,8,4,12,4,16,8,24,8,48,16,1,1,1,1,1,2,1,1,3,2,1,4, 2,1,6,2,1,8,4,1,12,4,1,16,8,1,24,8,1,48,16], [1,1,2,11,3,2,4,4,6,22,12,8,0,1,1,1,1,2,3,1,3,2,1,6,6,2,2,4,2,4,2,2,6, 8,2,12,4], [1,1,2,3,3,2,4,12,6,6,12,24,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,6,2,12,12], [1,1,2,7,3,2,4,8,6,14,12,16,0,1,1,1,1,2,1,1,3,2,1,4,2,1,6,2,1,12,4,2, 2,3,2,4,3,2,6,6,2,12,6], [1,1,2,7,3,2,4,8,6,14,12,16,0,1,1,1,1,2,3,1,3,2,1,6,6,2,2,2,2,4,4,2,6, 4,2,12,8], [1,1,2,3,3,2,4,12,6,6,12,24,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,6,2,12,12], [1,1,2,3,3,2,4,4,6,6,8,8,12,8,24,16,0,1,1,1,1,2,1,1,3,2,1,4,2,1,6,2, 1,12,4,2,2,1,2,4,1,2,6,2,2,8,4,2,12,2,2,24,8], [1,1,2,9,3,2,4,2,6,18,8,4,12,4,24,8,0,1,1,1,1,2,1,1,3,2,1,6,2,2,4,1, 2,8,2,2,12,2,2,24,4,4,2,2,4,6,4], [1,1,2,5,3,2,4,6,6,10,8,4,12,12,24,8,0,1,1,1,1,2,1,1,3,2,1,6,2,2,4,1, 2,8,2,2,12,2,2,24,4,4,2,1,4,4,1,4,6,2,4,12,2], [1,1,2,1,3,2,4,10,6,2,8,4,12,20,24,8,0,1,1,1,1,2,1,1,3,2,1,6,2,2,4,1, 2,8,2,2,12,2,2,24,4,4,4,2,4,12,4], [1,1,2,31,3,2,6,14,0,1,1,1,1,2,7,2,3,1,2,6,7,3,2,8], [1,1,2,15,3,2,4,16,6,6,12,8,0,1,1,1,1,2,3,1,4,4,2,3,1,2,6,3,2,12,4,3, 2,4,3,4,4], [1,1,2,7,3,2,4,24,6,14,0,1,1,1,1,2,7,2,3,1,2,6,7,3,4,8], [1,1,2,3,3,2,4,28,6,6,12,8,0,1,1,1,1,2,3,1,4,4,2,3,1,2,6,3,2,12,4,3,4,8], [1,1,2,7,3,2,4,8,6,2,8,16,12,4,24,8,0,1,1,1,1,2,1,1,4,2,1,8,4,2,3,1,2,6, 1,2,12,2,2,24,4,3,2,2,3,4,2,3,8,4], [1,1,2,3,3,2,4,4,6,6,8,24,12,8,0,1,1,1,1,2,3,1,4,4,2,3,1,2,6,3,2,12,4,3, 8,8], [1,1,2,1,3,2,4,2,6,2,8,4,12,4,16,24,24,8,0,1,1,1,1,2,1,1,4,2,1,8,4,2,3, 1,2,6,1,2,12,2,2,24,4,3,16,8], [1,1,2,15,3,8,6,24,0,1,1,1,1,2,3,3,2,4,4,3,2,4,6,6], [1,1,2,7,3,8,4,8,6,8,12,16,0,1,1,1,1,2,1,1,4,2,3,2,2,3,4,2,4,3,2,4,6,2, 4,12,4], [1,1,2,19,3,2,4,12,6,14,0,1,1,1,1,2,3,2,2,2,2,3,1,2,6,7,6,2,2,6,4,2], [1,1,2,27,3,2,4,4,6,6,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,12,4,6,2,4], [1,1,2,3,3,2,4,28,6,6,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,12,4,6,4,4], [1,1,2,7,3,2,4,24,6,14,0,1,1,1,1,2,3,2,2,2,2,3,1,2,6,7,6,4,4], [1,1,2,15,3,2,4,16,6,6,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,12,4,6,2, 2,6,4,2], [1,1,2,3,3,2,4,28,6,6,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,12,4,6,4,4], [1,1,2,3,3,2,4,28,6,6,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,12,4,6,4,4], [1,1,2,15,3,2,4,16,6,6,12,8,0,1,1,1,1,2,1,1,4,2,2,2,1,2,3,1,2,4,1,2,6,3, 2,12,4,6,2,2,6,4,2], [1,1,2,3,3,2,4,4,6,6,8,24,12,8,0,1,1,1,1,2,1,1,4,2,2,2,1,2,3,1,2,4,1,2,6, 3,2,12,4,6,8,4], [1,1,2,7,3,2,4,8,6,2,8,16,12,4,24,8,0,1,1,1,1,2,1,1,4,2,2,3,1,2,6,1,2,8, 2,2,12,2,2,24,4,6,2,1,6,4,1,6,8,2], [1,1,2,3,3,8,4,12,6,24,0,1,1,1,1,2,3,4,3,2,4,6,6,6,4,2], [1,1,2,7,3,8,4,8,6,8,12,16,0,1,1,1,1,2,1,1,4,2,4,3,2,4,6,2,4,12,4,6,2,1, 6,4,1], [1,1,2,19,3,8,4,12,6,8,0,1,1,1,1,2,1,3,2,2,6,2,2,6,4,2,8,3,1,8,6,1], [1,1,2,7,3,8,4,24,6,8,0,1,1,1,1,2,1,3,2,2,6,4,4,8,3,1,8,6,1], [1,1,2,23,3,2,4,8,6,10,12,4,0,1,1,1,1,2,1,2,2,2,2,3,1,2,4,1,2,6,1,3,2,2, 4,6,2,4,12,1,6,2,2,6,4,1], [1,1,2,7,3,2,4,24,6,2,12,12,0,1,1,1,1,2,1,2,3,1,2,4,3,2,6,1,3,2,2,4,12, 3,6,4,3], [1,1,2,11,3,2,4,20,6,10,12,4,0,1,1,1,1,2,1,2,2,2,2,3,1,2,4,1,2,6,1,3,4, 2,4,6,2,4,12,1,6,2,1,6,4,2], [1,1,2,19,3,2,4,12,6,2,12,12,0,1,1,1,1,2,1,2,3,1,2,4,3,2,6,1,3,4,2,4,12, 3,6,2,3], [1,1,2,25,3,2,4,2,6,2,8,4,12,4,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8, 2,2,12,2,2,24,4,12,2,2], [1,1,2,13,3,2,4,14,6,2,8,4,12,4,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2, 8,2,2,12,2,2,24,4,12,2,1,12,4,1], [1,1,2,1,3,2,4,26,6,2,8,4,12,4,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8, 2,2,12,2,2,24,4,12,4,2], [1,1,2,17,3,2,4,2,6,10,8,12,12,4,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,4,2,1, 4,6,2,4,12,1,6,8,2,12,2,1], [1,1,2,5,3,2,4,14,6,10,8,12,12,4,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,4,2,1, 4,6,2,4,12,1,6,8,2,12,4,1], [1,1,2,13,3,2,4,6,6,2,8,12,12,12,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,4,4,1, 4,12,3,6,8,2,12,2,1], [1,1,2,1,3,2,4,18,6,2,8,12,12,12,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,4,4,1, 4,12,3,6,8,2,12,4,1], [1,1,2,13,3,8,4,6,6,8,8,12,0,1,1,1,1,2,1,6,4,1,6,8,2,8,3,1,8,6,1,12,2,1], [1,1,2,1,3,8,4,18,6,8,8,12,0,1,1,1,1,2,1,6,4,1,6,8,2,8,3,1,8,6,1,12,4,1], [1,1,2,15,3,32,0,1,1,1,3,2,5,16,3,2], [1,1,2,3,3,32,4,12,0,1,1,1,3,2,1,3,4,4,16,3,2]]; ############################################################################# ## #F GroupIdOps.Group50( ) . . . . . . . . . . . . . . groups of order 50 ## GroupIdOps.Group50 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList50, x ); # and return return rec( catalogue := [ 50, nr ], names := GroupIdOps.GroupName50[nr], size := 50 ); end; GroupIdOps.GroupName50 := [ [ "10x5" ], [ "50" ], [ "D50" ], [ "D10x5" ], [ "5^2:2" ] ]; GroupIdOps.GroupList50 := [ [ 1, 1, 2, 1, 5, 24, 10, 24 ], [ 1, 1, 2, 1, 5, 4, 10, 4, 25, 20, 50, 20 ], [ 1, 1, 2, 25, 5, 4, 25, 20 ], [ 1, 1, 2, 5, 5, 24, 10, 20 ], [ 1, 1, 2, 25, 5, 24 ] ]; ############################################################################# ## #F GroupIdOps.Group54( ) . . . . . . . . . . . . . . groups of order 54 ## GroupIdOps.Group54 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList54, x ); # element orders for tupless 1/4, 2/5, 9/12, 8/10/11 are equal if nr = 1 and not IsAbelian(G) then nr := 4; fi; if nr = 2 and not IsAbelian(G) then nr := 5; fi; if nr = 9 and Length(Orbits(G,Elements(G))) = 10 then nr := 12; fi; if nr = 8 and Length(Orbits(G,Elements(G))) = 10 then if Length(DerivedSeries(G)) = 4 then nr := 10; else nr := 11; fi; fi; # and return return rec( catalogue := [ 54, nr ], names := GroupIdOps.GroupName54[nr], size := 54 ); end; GroupIdOps.GroupName54 := [ ["3^3x2"], ["9x6"], ["54"], ["3^2:3x2"], ["9:3x2"], ["3^2xS3"], ["9xS3"], ["3x3^2:2"], ["3xD18"], ["(3^2:3):2"], ["3^2:6"], ["9:6"], ["3^3:2"], ["(3x9):2"], ["D54"] ]; GroupIdOps.GroupList54 := [ [ 1, 1, 2, 1, 3, 26, 6, 26 ], [ 1, 1, 2, 1, 3, 8, 6, 8, 9, 18, 18, 18 ], [ 1, 1, 2, 1, 3, 2, 6, 2, 9, 6, 18, 6, 27, 18, 54, 18 ], [ 1, 1, 2, 1, 3, 26, 6, 26 ], [ 1, 1, 2, 1, 3, 8, 6, 8, 9, 18, 18, 18 ], [ 1, 1, 2, 3, 3, 26, 6, 24 ], # swapped: ** and ** (26,24), see catalogue [ 1, 1, 2, 3, 3, 8, 6, 6, 9, 18, 18, 18 ], [ 1, 1, 2, 9, 3, 26, 6, 18 ], [ 1, 1, 2, 9, 3, 8, 6, 18, 9, 18 ], [ 1, 1, 2, 9, 3, 26, 6, 18 ], [ 1, 1, 2, 9, 3, 26, 6, 18 ], [ 1, 1, 2, 9, 3, 8, 6, 18, 9, 18 ], [ 1, 1, 2, 27, 3, 26 ], [ 1, 1, 2, 27, 3, 8, 9, 18 ], [ 1, 1, 2, 27, 3, 2, 9, 6, 27, 18 ] ]; ############################################################################# ## #F GroupIdOps.Group56( ) . . . . . . . . . . . . . . groups of order 56 ## GroupIdOps.Group56 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList56, x ); # and return return rec( catalogue := [ 56, nr ], names := GroupIdOps.GroupName56[nr], size := 56 ); end; GroupIdOps.GroupName56 := [ ["2^3x7"], ["2x28"], ["56"], ["7xD8"], ["7xQ8"], ["2xD28"], ["4xD14"], ["2x14.2"], ["28.2"], [], ["D56"], ["Q8+D14"], ["2^3:7"] ]; GroupIdOps.GroupList56 := [ [ 1, 1, 2, 7, 7, 6, 14, 42 ], [ 1, 1, 2, 3, 4, 4, 7, 6, 14, 18, 28, 24 ], [ 1, 1, 2, 1, 4, 2, 7, 6, 8, 4, 14, 6, 28, 12, 56, 24 ], [ 1, 1, 2, 5, 4, 2, 7, 6, 14, 30, 28, 12 ], [ 1, 1, 2, 1, 4, 6, 7, 6, 14, 6, 28, 36 ], [ 1, 1, 2, 31, 7, 6, 14, 18 ], [ 1, 1, 2, 15, 4, 16, 7, 6, 14, 6, 28, 12 ], [ 1, 1, 2, 3, 4, 28, 7, 6, 14, 18 ], [ 1, 1, 2, 1, 4, 2, 7, 6, 8, 28, 14, 6, 28, 12 ], [ 1, 1, 2, 17, 4, 14, 7, 6, 14, 18 ], [ 1, 1, 2, 29, 4, 2, 7, 6, 14, 6, 28, 12 ], [ 1, 1, 2, 1, 4, 30, 7, 6, 14, 6, 28, 12 ], [ 1, 1, 2, 7, 7, 48 ] ]; ############################################################################# ## #F GroupIdOps.Group60( ) . . . . . . . . . . . . . . groups of order 60 ## GroupIdOps.Group60 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList60, x ); # and return return rec( catalogue := [ 60, nr ], names := GroupIdOps.GroupName60[nr], size := 60 ); end; GroupIdOps.GroupName60 := [ ["2^2x15"], ["60"], ["10xS3"], ["5x3:4"], ["5xA4"], ["6xD10"], ["3x10.2"], ["3x5:4"], ["D60"], ["30.2"], ["S3xD10"], ["S3+(5:4)"], ["A5","PSL(2,4)","PSL(2,5)"] ]; GroupIdOps.GroupList60 := [ [ 1, 1, 2, 3, 3, 2, 5, 4, 6, 6, 10, 12, 15, 8, 30, 24 ], [ 1, 1, 2, 1, 3, 2, 4, 2, 5, 4, 6, 2, 10, 4, 12, 4, 15, 8, 20, 8, 30, 8, 60, 16 ], [ 1, 1, 2, 7, 3, 2, 5, 4, 6, 2, 10, 28, 15, 8, 30, 8 ], [ 1, 1, 2, 1, 3, 2, 4, 6, 5, 4, 6, 2, 10, 4, 15, 8, 20, 24, 30, 8 ], [ 1, 1, 2, 3, 3, 8, 5, 4, 10, 12, 15, 32 ], [ 1, 1, 2, 11, 3, 2, 5, 4, 6, 22, 10, 4, 15, 8, 30, 8 ], [ 1, 1, 2, 1, 3, 2, 4, 10, 5, 4, 6, 2, 10, 4, 12, 20, 15, 8, 30, 8 ], [ 1, 1, 2, 5, 3, 2, 4, 10, 5, 4, 6, 10, 12, 20, 15, 8 ], [ 1, 1, 2, 31, 3, 2, 5, 4, 6, 2, 10, 4, 15, 8, 30, 8 ], [ 1, 1, 2, 1, 3, 2, 4, 30, 5, 4, 6, 2, 10, 4, 15, 8, 30, 8 ], [ 1, 1, 2, 23, 3, 2, 5, 4, 6, 10, 10, 12, 15, 8 ], [ 1, 1, 2, 5, 3, 2, 4, 30, 5, 4, 6, 10, 15, 8 ], [ 1, 1, 2, 15, 3, 20, 5, 24 ] ]; ############################################################################# ## #F GroupIdOps.Group64( ) . . . . . . . . . . . . . . groups of order 64 ## GroupIdOps.Group64 := function( G ) local z1, z2, z3, z4, z5, nr; # compute element orders z1 := Collected( List( Elements(G), x -> Order(G,x) ) ); # abelian or nor abelian? if IsAbelian(G) then z2 := 0; else z2 := -1; fi; # conjugacy classes z3 := List( Orbits( G, Elements(G) ), x -> [Length(x),Order(G,x[1])] ); Sort(z3); # and squares z4 := GroupIdOps.Nr4Squares(G); z5 := GroupIdOps.Nr8Squares(G); # fingerprint z1 := Flat( Concatenation( z1, [z2], z3, z4, z5 ) ); # look it up nr := Position( GroupIdOps.GroupList64, z1 ); # handle various problem cases if nr = 23 then if GroupIdOps.Stable( G, 8, 5 ) = 0 then nr := 37; fi; elif nr = 26 then if GroupIdOps.Stable( G, 4, 3 ) = 0 then nr := 40; fi; elif nr = 31 then if GroupIdOps.Stable( G, 8, 5 ) = 0 then nr := 33; fi; elif nr = 39 then if GroupIdOps.Stable( G, 4, 3 ) = 2 then nr := 41; fi; elif nr = 50 then if GroupIdOps.Stable( G, 8, 3 ) = 0 then nr := 59; fi; elif nr = 87 then if Length(ConjugacyClassesSubgroups(G)) = 85 then nr := 88; fi; elif nr = 90 then if GroupIdOps.Stable2(G) = [ 2, 2, 4, 6, 6 ] then nr := 92; fi; elif nr = 115 then if GroupIdOps.Stable( G, 4, 3 ) = 4 then nr := 116; fi; elif nr = 147 then if Length(ConjugacyClassesSubgroups(G)) = 93 then nr := 148; fi; elif nr = 165 then if GroupIdOps.Stable2(G) = [ 5, 6, 6 ] then nr := 166; fi; elif nr = 183 then if Length(ConjugacyClassesSubgroups(G)) = 126 then nr := 184; fi; elif nr = 189 then z1 := GroupIdOps.Stable( G, 8, 3 ); if z1 = 0 then nr := 193; elif z1 = 4 then nr := 198; fi; elif nr = 190 then if Length(ConjugacyClassesSubgroups(G)) = 61 then nr := 192; fi; elif nr = 191 then z1 := GroupIdOps.Stable( G, 8, 3 ); if z1 = 0 then nr := 194; elif z1 = 4 then nr := 199; fi; elif nr = 203 then if GroupIdOps.Stable( G, 4, 3 ) = 4 then nr := 213; fi; elif nr = 208 then z1 := GroupIdOps.Stable2(G); if z1 = [ 2, 3, 4 ] then nr := 209; elif z1 = [ 2, 2, 5 ] then nr := 221; fi; elif nr = 211 then if GroupIdOps.Stable2(G) = [ 2, 4, 5 ] then nr := 212; fi; elif nr = 210 then if GroupIdOps.Stable( G, 4, 3 ) = 7 then nr := 222; fi; elif nr = 235 then if GroupIdOps.Stable2(G) = [2,2,3] then nr := 236; fi; fi; # and return return rec( catalogue := [ 64, nr ], names := GroupIdOps.GroupName64[nr], size := 64 ); end; GroupIdOps.GroupName64 := [ ["2^6"],["2^4x4"],["2^3x8"],["4^2x2^2"],["2^2x16"],["2x4x8"],["2x32"], ["4^3"],["4x16"],["8^2"],["64"],["2^3xD8"],["2^3xQ8"],["2^2xD8Y4"],[], [],[],["2x4xD8"],[],[],[],[],[],[],[],[],["4xD8Y4"],[],[],[],[],[],[], ["8xD8"],["8xQ8"],["16YD8"],[],[],[],[],[],[],["2^2xD16"],["2^2xQD16"], ["2^2xQ16"],["2xD16Y4"],[],[],[],[],[],[],[],[],["4xD16"],["4xQD16"], ["4xQ16"],["8YD16"],[],[],[],[],[],[],[],[],[],[],[],["2xQ8+Q8"],[],[], [],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[], [],[],[],[],[],[],[],["2xD8YD8"],["2xD8YQ8"],[],[],[],[],[],[],[],[], [],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[], ["2xD32"],["2xQD32"],["2xQ32"],[],[],[],[],[],[],[],[],[],[],[],[],[], [],[],[],[],["D8xD8"],["D8xQ8"],["Q8xQ8"],[],[],[],[],[],[],[],[],[], [],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[], [],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[], [],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[], [],[],[],[],[],[],["D8YD16"],["D8YQD16"],["Q8YD16"],[],[],[],[],[],[], [],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[]]; GroupIdOps.GroupList64 := [ [1,1,2,63,0,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2],[1,1,2,31,4,32,0,1, 1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1, 2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,32],[1,1,2,15,4,16,8,32,0,1,1,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8, 1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8, 1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,16,16,16],[1,1,2,15,4,48,0,1,1,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,16,16,16],[1,1,2,7,4,8,8,16,16,32,0,1,1,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8, 1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,16,1,16,1,16,1,16,1,16, 1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1, 16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,8,8,8],[1, 1,2,7,4,24,8,32,0,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1, 4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1, 8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,8,8,8,8,8,8,8], [1,1,2,3,4,4,8,8,16,16,32,32,0,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,8,1,8, 1,8,1,8,1,8,1,8,1,8,1,8,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1, 16,1,16,1,16,1,16,1,16,1,16,1,16,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32, 1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1, 32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,4,4,4],[1,1,2,7,4,56,0, 1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,8,8,8,8,8,8,8],[1,1,2,3,4,12,8, 16,16,32,0,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4, 1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,16, 1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1, 16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16, 1,16,1,16,4,4,4,4,4,4,4],[1,1,2,3,4,12,8,48,0,1,1,1,2,1,2,1,2,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1, 8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1, 8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1, 8,1,8,1,8,1,8,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4],[1,1,2,1,4,2,8,4,16,8,32,16, 64,32,0,1,1,1,2,1,4,1,4,1,8,1,8,1,8,1,8,1,16,1,16,1,16,1,16,1,16,1,16,1, 16,1,16,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32,1,32, 1,32,1,32,1,32,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1, 64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64,1,64, 1,64,1,64,1,64,1,64,1,64,1,64,2,2,2],[1,1,2,47,4,16,-1,1,1,1,2,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 16],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1, 2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,48],[1,1,2,31,4,32,-1,1,1,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 2,4,32],[1,1,2,31,4,32,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1, 2,1,2,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,16,16],[1,1,2,15,4,48,-1,1,1, 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 2,4,2,4,2,4,16,32],[1,1,2,15,4,16,8,32,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1, 2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,8,2, 8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,16,16,16],[1, 1,2,23,4,40,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4, 1,4,1,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,8,8,24],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1, 2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,8,24,24],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,1,4,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,16,16,16],[1,1,2,15,4,16, 8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8, 2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,16,16,16],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,16, 16,16],[1,1,2,7,4,24,8,32,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2, 8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,8,8,8,8,8,8],[1,1,2,15, 4,16,8,32,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1, 4,1,4,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2, 8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,16,8,8,8,8],[1,1,2,7,4,24,8,32,-1,1,1,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,8,8,8,16,16],[1,1,2,7,4,8,8,16,16,32,-1,1,1,1,2,1,2,1,2,1,4,1,4, 1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,2,2,2,2,2,4,2,4,2,8,2,8,2,8,2,8, 2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2, 16,2,16,8,8,8],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,1,4,1,4,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,16,16,16],[1,1,2,15,4, 48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2, 2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,8,8,8,8,16],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2, 1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,8,8, 8,16,16],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1, 4,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,8,8,8,8,8,8,8],[1,1,2,7,4,24, 8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4, 2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,8,8,8,8,8,8,8],[1,1,2,7,4,24,8,32,-1,1,1,1,2,1, 2,1,2,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,2,2,2,2,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2, 8,2,8,8,8,8,8,8,8,8],[1,1,2,7,4,24,8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,8,8,8,8,8, 8],[1,1,2,11,4,20,8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1, 8,1,8,1,8,1,8,1,8,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2, 8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,4,12,4,4,12,12],[1,1,2,3,4, 28,8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8, 1,8,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,12,12,4,4,12,12],[1,1,2,7,4,8,8,16,16,32, -1,1,1,1,2,1,4,1,4,1,8,1,8,1,8,1,8,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1, 16,2,2,2,2,2,2,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,16,2,16,2,16,2,16, 2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,8,8,8],[1,1,2,7,4,24,8,32,-1,1, 1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2, 8,2,8,2,8,2,8,8,8,8,8,8,8,8],[1,1,2,3,4,12,8,48,-1,1,1,1,2,1,2,1,2,1,4, 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4],[1,1,2,3,4,12,8,16,16,32,-1,1,1,1,2,1,2,1,2, 1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,2,4,2,4,2,4,2,4,2,8,2,8, 2,8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16, 2,16,2,16,2,16,4,4,4,4,4,4,4],[1,1,2,7,4,8,8,16,16,32,-1,1,1,1,2,1,2,1, 2,1,4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,2,2,2,2,2,4,2,4,2,8,2, 8,2,8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2, 16,2,16,2,16,2,16,8,8,8],[1,1,2,3,4,12,8,16,16,32,-1,1,1,1,2,1,2,1,2,1, 4,1,4,1,4,1,4,1,8,1,8,1,8,1,8,1,8,1,8,1,8,1,8,2,4,2,4,2,4,2,4,2,8,2,8,2, 8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16, 2,16,2,16,2,16,4,4,4,4,4,4,4],[1,1,2,3,4,4,8,8,16,16,32,32,-1,1,1,1,2,1, 4,1,4,1,8,1,8,1,8,1,8,1,16,1,16,1,16,1,16,1,16,1,16,1,16,1,16,2,2,2,4,2, 8,2,8,2,16,2,16,2,16,2,16,2,32,2,32,2,32,2,32,2,32,2,32,2,32,2,32,2,32, 2,32,2,32,2,32,2,32,2,32,2,32,2,32,4,4,4],[1,1,2,39,4,8,8,16,-1,1,1,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,8,8,8],[1,1,2,23,4,24,8,16,-1,1,1,1,2,1, 2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4, 2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,24,8,8],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,40,8,8],[1,1,2,23,4,24,8,16,-1,1,1,1,2,1,2, 1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,2, 4,2,4,2,4,2,4,4,4,4,4,4,4,4,24,8,8],[1,1,2,23,4,24,8,16,-1,1,1,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,2, 4,2,4,2,4,2,4,4,4,4,4,4,4,4,8,16,8,8],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,16,24,8,8],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1, 2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,32,8,8],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1, 2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,16,16,8,8],[1,1,2,15,4,32,8,16,-1,1,1,1, 2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,4,2,4,2,4,4,4,4,4,8,4,8,4,8,4,8,8,8,16,8,8],[1,1,2,7,4,8,8,48,-1,1,1, 1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,8,8,8,16,16],[1,1,2,15,4,32,8,16,-1, 1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,8,16,8,8],[1,1,2,7,4,40,8,16, -1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,16,16,8,8],[1,1,2,19,4,28, 8,16,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,2,8,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,20,4,4,4,4],[1,1,2, 11,4,36,8,16,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,12,12,12,4,4,4, 4],[1,1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2, 4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,20, 20,4,4,4,4],[1,1,2,11,4,20,8,32,-1,1,1,1,2,1,4,1,4,1,8,1,8,1,8,1,8,2,2, 2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,4,2,4,2,4,4,4,4,4,8,4,8,4,8, 4,8,4,4,12,4,4,12,12],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2, 1,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,8,16,16,8,8],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2,1,4,1,4, 1,4,1,4,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4, 4,8,4,8,4,8,4,8,8,8,8,8,8,8,8],[1,1,2,7,4,8,8,48,-1,1,1,1,2,1,2,1,2,1,4, 1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,8,4,8,4,8, 4,8,4,8,4,8,4,8,4,8,8,8,8,8,8,8,8],[1,1,2,11,4,20,8,32,-1,1,1,1,2,1,2,1, 2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,2,4, 2,4,4,4,4,4,8,4,8,4,8,4,8,4,4,12,4,4,4,4,8,8],[1,1,2,3,4,28,8,32,-1,1,1, 1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,4,12,12,4,4,4,4,8,8],[1,1,2,3,4,12, 8,48,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,2,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,4,4,4,4,4,4,16,16],[1, 1,2,3,4,12,8,48,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,4,4,4,4,4, 4,8,8,8,8],[1,1,2,7,4,8,8,16,16,32,-1,1,1,1,2,1,4,1,4,1,8,1,8,1,8,1,8,2, 2,2,4,2,8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,4,2,4,4,4,8,4,8,4, 16,4,16,4,16,4,16,8,8,8],[1,1,2,3,4,12,8,16,16,32,-1,1,1,1,2,1,4,1,4,1, 8,1,8,1,8,1,8,2,2,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,4,16,4,16, 4,16,4,16,4,16,4,16,4,16,4,16,4,4,4,4,4,4,4],[1,1,2,39,4,24,-1,1,1,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, 4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8],[1,1,2,39,4,24,-1,1,1,1,2,1,2,1, 2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,8,8,8],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1, 2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,8,8,40],[1,1,2,31,4,32,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2, 1,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4, 4,4,4,4,4,4,16,16],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2, 2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,16,32],[1,1,2,23,4,40,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,2,2,2, 2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4, 8,8,24],[1,1,2,23,4,40,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,8,8,24], [1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,24,24],[1,1, 2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,16,16,16],[1,1,2, 23,4,40,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,2,2,2,2,2,2,2,2,4, 2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,8,24],[1,1,2,23,4, 40,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,8,8,24],[1,1,2,7,4,56,-1, 1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,24,24],[1,1,2,15,4,48,-1,1,1, 1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,16,16,16],[1,1,2,31,4,32,-1,1,1,1,2, 1,2,1,2,1,2,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,8],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1, 2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,32],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1, 2,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,16,16,16],[1,1,2,23,4,40,-1,1,1,1,2,1,2,1,2,1,2,1, 2,1,2,1,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4, 4,4,4,4,4,4,4,4,4,8,8,8,16],[1,1,2,23,4,40,-1,1,1,1,2,1,2,1,2,1,2,1,2,1, 2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4, 4,4,4,4,4,4,4,8,8,8,16],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1, 2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,8,8,8,24],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,8,8,16,24],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,8,8,16,24],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,2,2,2, 2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 8,8,8,8,16],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8, 8,8,16,16],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8, 8,8,8,16],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8, 8,16,16],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8, 8,8,8,8],[1,1,2,15,4,16,8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2,2, 2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8, 16,8,8,8,8],[1,1,2,7,4,24,8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,2,2, 2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4, 8,8,8,8,16,16],[1,1,2,15,4,16,8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2, 2,2,2,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,2,4,2,4,4,4,4,4,8,4,8,4, 8,4,8,16,8,8,8,8],[1,1,2,7,4,24,8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4, 2,2,2,2,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,4,4,4,4,4,4,4,4,8,4,8, 4,8,4,8,8,8,8,16,16],[1,1,2,7,4,24,8,32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4, 1,4,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,8,4,8,4,8,4,8,4,8, 4,8,4,8,4,8,8,8,8,8,8,8,8],[1,1,2,11,4,20,8,32,-1,1,1,1,2,1,2,1,2,1,4,1, 4,1,4,1,4,2,2,2,2,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,2,4,4,4,4,4, 4,4,8,4,8,4,8,4,8,4,4,12,4,4,12,12],[1,1,2,11,4,20,8,32,-1,1,1,1,2,1,2, 1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,4,2, 4,2,4,4,4,4,4,8,4,8,4,8,4,8,4,4,12,4,4,12,12],[1,1,2,3,4,28,8,32,-1,1,1, 1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,4,12,12,4,4,12,12],[1,1,2,7,4,24,8, 32,-1,1,1,1,2,1,2,1,2,1,4,1,4,1,4,1,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2, 8,2,8,2,8,2,8,4,2,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,8,8,8,8,8,8],[1,1,2,39, 4,24,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 24],[1,1,2,23,4,40,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, 2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,40],[1,1,2,31,4,32,-1,1,1,1,2,1,4,1,4,2,2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,2,4,32],[1,1,2,23,4,40,-1,1,1,1,2,1,2,1,2,2,2,2, 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,8,16,16],[1,1,2,15,4,48,-1,1, 1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,8,16,24], [1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,16,16,24],[1,1,2,15,4,16,8,32,-1,1,1,1,2,1,4,1,4,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,16,16,16],[1,1,2,31,4,16,8,16,-1,1, 1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4, 8,4,8,4,8,4,8,16,8,8],[1,1,2,15,4,32,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2, 4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,32,8,8],[1, 1,2,23,4,24,8,16,-1,1,1,1,2,1,4,1,4,2,2,2,2,2,2,2,4,2,4,2,4,4,2,4,2,4,2, 4,2,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,24,8,8],[1,1,2,23,4,24,8,16,-1,1,1, 1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,8, 4,8,4,8,4,8,8,8,8,8,8],[1,1,2,15,4,32,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2, 4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,16,16,8,8], [1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,8,24,8,8],[1,1,2,7,4,40,8,16,-1, 1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,8,4,8,4,8,4,8,8,8,24,8,8],[1,1,2,19,4,28,8,16,-1,1,1,1,2,1,2,1,2,2,4, 2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,4,12, 12,4,4,4,4],[1,1,2,11,4,36,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4, 2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,4,12,20,4,4,4,4],[1, 1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,12,12,20,4,4,4,4],[1,1,2,15,4,32,8, 16,-1,1,1,1,2,1,4,1,4,2,2,2,2,2,2,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4, 4,4,4,4,8,4,8,4,8,4,8,8,8,16,8,8],[1,1,2,7,4,8,8,48,-1,1,1,1,2,1,4,1,4, 2,2,2,2,2,2,2,4,2,4,2,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8, 8,8,8,16,16],[1,1,2,11,4,20,8,32,-1,1,1,1,2,1,4,1,4,2,2,2,4,2,8,2,8,2,8, 2,8,4,2,4,2,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,4,8,4,8,4,4,12,4,4,12,12],[1, 1,2,23,4,40,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,8,8],[1,1,2,23,4,8,8,32,-1,1,1,1, 2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,8,4,8,4,8,4,8,4,8,4, 8,4,8,4,8,8,8,8,8,8],[1,1,2,7,4,24,8,32,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4, 2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,24,8,8,8,8], [1,1,2,15,4,48,-1,1,1,1,2,1,4,1,4,2,2,2,2,2,2,2,4,2,4,2,4,4,2,4,2,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,8,16],[1,1,2,15,4,16,8,32,-1,1, 1,1,2,1,4,1,4,2,2,2,2,2,2,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,8,4,8,4,8,4,8,4, 8,4,8,4,8,4,8,16,8,8,8,8],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,2, 2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,8,8,8], [1,1,2,7,4,24,8,32,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,4,4,4,4, 4,4,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,8,16,8,8,8,8],[1,1,2,7,4,24,8,32, -1,1,1,1,2,1,4,1,4,2,2,2,2,2,2,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8, 4,8,4,8,4,8,4,8,4,8,8,8,8,8,8,8,8],[1,1,2,15,4,16,8,32,-1,1,1,1,2,1,2,1, 2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4, 8,8,8,4,4,4,4,4,4,4,4],[1,1,2,7,4,24,8,32,-1,1,1,1,2,1,2,1,2,2,2,2,2,2, 4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,8,16,4,4, 4,4,4,4,4,4],[1,1,2,7,4,8,8,16,16,32,-1,1,1,1,2,1,4,1,4,2,2,2,4,2,8,2,8, 2,8,2,8,4,2,4,4,4,8,4,8,4,16,4,16,4,16,4,16,4,16,4,16,4,16,4,16,8,8,8], [1,1,2,35,4,4,8,8,16,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,2,16, 2,16,2,16,2,16,2,16,2,16,2,16,2,16,8,2,8,2,8,2,8,2,4,4,4],[1,1,2,19,4,20, 8,8,16,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,2,16,2,16,2,16,2,16, 2,16,2,16,2,16,2,16,8,2,8,2,8,4,8,4,20,4,4],[1,1,2,3,4,36,8,8,16,16,-1, 1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2, 16,2,16,8,4,8,4,8,4,8,4,36,4,4],[1,1,2,19,4,20,8,8,16,16,-1,1,1,1,2,1,4, 1,4,2,2,2,4,2,8,2,8,2,8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,8,2, 8,2,8,4,8,4,20,4,4],[1,1,2,19,4,20,8,8,16,16,-1,1,1,1,2,1,2,1,2,2,4,2,4, 2,8,2,8,2,8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,8,2,8,2,8,4,8,4, 4,16,4,4],[1,1,2,3,4,36,8,8,16,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2, 8,2,8,2,16,2,16,2,16,2,16,2,16,2,16,2,16,2,16,8,4,8,4,8,4,8,4,16,20,4,4], [1,1,2,3,4,36,8,8,16,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,2,16, 2,16,2,16,2,16,2,16,2,16,2,16,2,16,8,4,8,4,8,4,8,4,4,32,4,4],[1,1,2,3,4, 36,8,8,16,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,8,2,8,2,8,2,8,2,16,2,16,2,16, 2,16,2,16,2,16,2,16,2,16,8,4,8,4,8,4,8,4,4,16,16,4,4],[1,1,2,11,4,12,8, 24,16,16,-1,1,1,1,2,1,4,1,4,2,2,2,4,2,8,2,8,2,8,2,8,2,16,2,16,2,16,2,16, 2,16,2,16,2,16,2,16,8,2,8,4,8,8,8,8,12,4,4,8,8],[1,1,2,3,4,4,8,40,16,16, -1,1,1,1,2,1,4,1,4,2,2,2,4,2,8,2,8,2,8,2,8,2,16,2,16,2,16,2,16,2,16,2,16, 2,16,2,16,8,8,8,8,8,8,8,8,4,4,4,16,16],[1,1,2,31,4,32,-1,1,1,1,2,1,2,1, 2,1,2,1,2,1,2,1,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,8,8,8,8],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,32],[1,1,2,23,4,40,-1, 1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,8,8,8,16],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2, 1,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,24],[1, 1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,4,2,4,2,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,24],[1,1,2,7,4,56,-1,1,1,1,2,1,2, 1,2,1,2,1,2,1,2,1,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,8,8,16,24],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,4,2,4, 2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,16,16,16],[1,1,2,15,4, 48,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,8,8,8,8,16],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2, 1,2,1,2,1,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8, 8,16,16],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,8,8,8,8],[1,1,2,35,4,28, -1,1,1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2, 4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,12,12],[1,1,2,11,4,52,-1,1,1,1,2,1,2, 1,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,12,36],[1,1,2,3,4,60,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,12, 12,36],[1,1,2,27,4,36,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2, 4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,12,20],[1,1,2,27, 4,36,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4, 4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,12,20],[1,1,2,19,4,44,-1,1,1,1,2, 1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,4, 4,4,4,4,4,4,4,4,4,4,4,12,28],[1,1,2,19,4,44,-1,1,1,1,2,1,2,1,2,2,2,2,2, 2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,12,28],[1,1,2,11,4,52,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,4,2,4, 2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,20,28],[1, 1,2,3,4,60,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,12,20,28],[1,1,2,19,4,44,-1,1, 1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,12,12,20],[1,1,2,19,4,44,-1,1,1,1,2,1,2,1,2,2, 2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,4,4,4,4,4,4, 4,4,4,4,4,12,12,20],[1,1,2,19,4,44,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,12,12, 20],[1,1,2,19,4,44,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2, 4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,12,12,20],[1,1,2,11,4, 52,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4, 2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,12,20,20],[1,1,2,11,4,52,-1,1,1,1,2,1, 2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,12,20,20],[1,1,2,31,4,32,-1,1,1,1,2,1,2,1,2,2,2,2,2,2, 2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,16],[1, 1,2,31,4,32,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2, 4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,8,16],[1,1,2,23,4,40,-1,1,1,1,2,1,2,1, 2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,16,24],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2, 4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,32],[1,1,2,15,4,48,-1,1, 1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,24,24],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2, 4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,16,32],[1,1,2,23, 4,40,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,8,16,16],[1,1,2,23,4,40,-1,1,1,1,2,1,2,1,2,2,4, 2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,16, 16],[1,1,2,15,4,48,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,4,2,4,2,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,8,16,24],[1,1,2,15,4,48,-1,1,1,1, 2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,8,16,24],[1,1,2,7,4,56,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4, 2,4,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,16,16,24],[1,1,2,7, 4,40,8,16,-1,1,1,1,2,1,4,1,4,2,2,2,2,2,2,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,8,8,8,8,8,8],[1,1,2,7,4,8,8,48,-1,1,1,1, 2,1,4,1,4,2,2,2,2,2,2,2,4,2,4,2,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4, 8,4,8,4,8,8,8,8,8,8,8,8],[1,1,2,3,4,12,8,16,16,32,-1,1,1,1,2,1,4,1,4,2, 2,2,4,2,8,2,8,2,8,2,8,4,4,4,4,4,8,4,8,4,16,4,16,4,16,4,16,4,16,4,16,4,16, 4,16,4,4,4,4,4,4,4],[1,1,2,27,4,36,-1,1,1,1,2,1,2,1,2,4,2,4,2,4,2,4,2,4, 2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,12,12,12],[1,1,2,27,4,36,-1,1, 1,1,2,1,2,1,2,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,12,12,12],[1,1,2,19,4,44,-1,1,1,1,2,1,2,1,2,4,2,4,2,4,2,4,2,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,20,20],[1,1,2,11,4,52,-1,1,1,1,2,1, 2,1,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,12,12, 28],[1,1,2,3,4,60,-1,1,1,1,2,1,2,1,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,4,4,4,20,20,20],[1,1,2,35,4,12,8,16,-1,1,1,1,2,1,2, 1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,2,8,2,8,2, 8,2,4,4,4,4,4,4,4],[1,1,2,19,4,28,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2, 4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,2,8,2,8,4,8,4,4,4,20,4,4,4, 4],[1,1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2, 8,2,8,2,8,2,8,2,8,2,8,2,8,8,4,8,4,8,4,8,4,4,4,36,4,4,4,4],[1,1,2,3,4,44, 8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8, 2,8,2,8,8,4,8,4,8,4,8,4,4,20,20,4,4,4,4],[1,1,2,3,4,44,8,16,-1,1,1,1,2, 1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,4,8,4, 8,4,8,4,4,4,36,4,4,4,4],[1,1,2,19,4,28,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4, 2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,2,8,2,8,4,8,4,4,4,20, 4,4,4,4],[1,1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4, 2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,4,8,4,8,4,8,4,4,20,20,4,4,4,4],[1,1,2, 23,4,24,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8, 2,8,2,8,2,8,2,8,8,2,8,2,8,4,8,4,8,16,8,8],[1,1,2,15,4,32,8,16,-1,1,1,1, 2,1,2,1,2,2,2,2,2,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,2,8, 4,8,4,8,4,8,8,16,8,8],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4, 2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,4,8,4,8,4,8,4,16,24,8,8], [1,1,2,19,4,28,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8, 2,8,2,8,2,8,2,8,2,8,2,8,8,2,8,2,8,4,8,4,4,4,20,4,4,4,4],[1,1,2,3,4,44,8, 16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2, 8,2,8,8,4,8,4,8,4,8,4,4,20,20,4,4,4,4],[1,1,2,11,4,36,8,16,-1,1,1,1,2,1, 2,1,2,2,4,2,4,2,4,2,4,2,4,2,4,2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,8,2,8,4,8, 4,8,4,12,12,12,4,4,4,4],[1,1,2,31,4,16,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2, 2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,8,4,8,4,8,4,8,8,2,8,4,8,8,4,4,4,4],[1,1,2, 23,4,24,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,8, 4,8,4,8,4,8,8,4,8,4,8,16,4,4,4,4],[1,1,2,15,4,32,8,16,-1,1,1,1,2,1,2,1, 2,2,2,2,2,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8,4,8,24,4,4, 4,4],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,4,4,4,4,4,4, 4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,8,32,4,4,4,4],[1,1,2,27,4,20,8,16,-1,1, 1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8, 2,4,4,12,8,8],[1,1,2,19,4,28,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4, 4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8,2,4,12,12,8,8],[1,1,2,11,4,36,8, 16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,8,4,8,4,8,4, 8,8,4,8,4,4,4,28,8,8],[1,1,2,11,4,36,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2, 4,2,4,4,2,4,2,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,4,12,20,8,8],[1,1,2,11, 4,36,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,8,4, 8,4,8,4,8,8,4,8,4,4,12,20,8,8],[1,1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,2, 4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,12,12,20,8,8], [1,1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4, 4,4,8,4,8,4,8,4,8,8,4,8,4,4,12,28,8,8],[1,1,2,3,4,44,8,16,-1,1,1,1,2,1, 2,1,2,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,4,12, 28,8,8],[1,1,2,15,4,32,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,4,4,4,4, 4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8,4,8,24,4,4,4,4],[1,1,2,15,4,32,8,16,-1, 1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2, 8,4,8,8,16,4,4,4,4],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2, 4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,8,16,16,4,4,4,4],[1,1,2,7, 4,40,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,4, 8,4,8,4,8,8,4,8,4,16,24,4,4,4,4],[1,1,2,23,4,24,8,16,-1,1,1,1,2,1,2,1,2, 2,4,2,4,2,4,2,4,4,2,4,2,4,2,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8,4,8,16,4,4,4, 4],[1,1,2,15,4,32,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,2,4,2,4,2,4, 4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,8,24,4,4,4,4],[1,1,2,19,4,28,8,16,-1,1,1, 1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8,4, 4,12,12,8,8],[1,1,2,11,4,36,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,4, 4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8,4,4,12,20,8,8],[1,1,2,11,4,36,8,16, -1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,8,4,8,4,8,4,8, 8,4,8,4,4,12,20,8,8],[1,1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4, 2,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,12,12,20,8,8],[1,1,2,15, 4,32,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,2,4,4,4,4,4,4,4,4,4,8,4, 8,4,8,4,8,8,2,8,4,8,8,16,4,4,4,4],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2, 2,4,2,4,2,4,2,4,4,2,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,8,16,16,4,4, 4,4],[1,1,2,27,4,20,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,4,4,4,4,4,8,4,8,4,8, 4,8,8,2,8,2,8,2,8,4,4,4,12,4,4,4,4],[1,1,2,11,4,36,8,16,-1,1,1,1,2,1,2, 1,2,2,4,2,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8,4,8,4,8,4,4,4,28,4,4,4,4],[1, 1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,4,4,4,4,4,8,4,8,4,8,4,8,8,4, 8,4,8,4,8,4,4,12,28,4,4,4,4],[1,1,2,23,4,24,8,16,-1,1,1,1,2,1,2,1,2,2,4, 2,4,4,2,4,4,4,8,4,8,4,8,4,8,8,2,8,2,8,4,8,4,8,8,8,8,8],[1,1,2,15,4,32,8, 16,-1,1,1,1,2,1,2,1,2,2,4,2,4,4,2,4,4,4,8,4,8,4,8,4,8,8,2,8,4,8,4,8,4,16, 16,8,8],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,4,2,4,4,4,8,4,8,4, 8,4,8,8,4,8,4,8,4,8,4,8,8,24,8,8],[1,1,2,19,4,28,8,16,-1,1,1,1,2,1,2,1, 2,2,4,2,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8,2,8,4,8,4,4,12,12,4,4,4,4],[1,1, 2,11,4,36,8,16,-1,1,1,1,2,1,2,1,2,2,4,2,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8, 4,8,4,8,4,4,12,20,4,4,4,4],[1,1,2,3,4,44,8,16,-1,1,1,1,2,1,2,1,2,2,4,2, 4,4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,8,4,8,4,12,12,20,4,4,4,4],[1,1,2,19,4, 12,8,32,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,4,4,8,4,8,4,8,4,8,4,8,4,8, 4,8,4,8,8,2,8,2,4,4,4,8,8,8,8],[1,1,2,3,4,28,8,32,-1,1,1,1,2,1,2,1,2,2, 4,2,4,2,4,2,4,4,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,8,4,8,4,4,4,20,8,8,8, 8],[1,1,2,3,4,28,8,32,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,4,4,8,4,8,4, 8,4,8,4,8,4,8,4,8,4,8,8,4,8,4,4,4,20,8,8,8,8],[1,1,2,15,4,32,8,16,-1,1, 1,1,2,1,2,1,2,2,2,2,2,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,2,8, 4,8,8,8,8,4,4,4,4],[1,1,2,7,4,40,8,16,-1,1,1,1,2,1,2,1,2,2,2,2,2,2,4,2, 4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,8,8,4,8,4,8,8,8,16,4,4,4,4],[1,1,2, 11,4,20,8,32,-1,1,1,1,2,1,2,1,2,2,4,2,4,2,4,2,4,4,4,4,8,4,8,4,8,4,8,4,8, 4,8,4,8,4,8,8,2,8,4,4,4,12,4,4,4,4,4,4,4,4],[1,1,2,3,4,28,8,32,-1,1,1,1, 2,1,2,1,2,2,4,2,4,2,4,2,4,4,4,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,8,4,8,4,4, 12,12,4,4,4,4,4,4,4,4],[1,1,2,31,4,16,8,16,-1,1,1,1,2,2,2,2,2,2,2,2,4,2, 4,2,4,2,4,2,8,2,8,4,2,4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,8,4,8,4,8,16,8,8],[1, 1,2,23,4,24,8,16,-1,1,1,1,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,8,2,8,4,2,4,2, 4,2,4,2,4,4,4,4,4,4,4,4,4,8,4,8,4,8,24,8,8],[1,1,2,15,4,32,8,16,-1,1,1, 1,2,2,2,2,2,2,2,2,4,2,4,2,4,2,4,2,8,2,8,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4, 4,8,4,8,4,8,32,8,8],[1,1,2,27,4,12,8,8,16,16,-1,1,1,1,2,2,2,2,4,2,4,2,8, 2,8,4,8,4,16,4,16,4,16,4,16,8,2,8,2,8,2,8,4,12,4,4],[1,1,2,11,4,28,8,8, 16,16,-1,1,1,1,2,2,2,2,4,2,4,2,8,2,8,4,8,4,16,4,16,4,16,4,16,8,2,8,4,8, 4,8,4,28,4,4],[1,1,2,3,4,20,8,24,16,16,-1,1,1,1,2,2,2,2,4,2,4,2,8,2,8,4, 8,4,16,4,16,4,16,4,16,8,4,8,4,8,8,8,8,4,8,8,4,4,8,8],[1,1,2,19,4,4,8,24, 16,16,-1,1,1,1,2,2,2,2,4,2,4,2,8,2,8,4,8,4,16,4,16,4,16,4,16,8,2,8,2,8, 8,8,8,4,4,4,8,8],[1,1,2,11,4,28,8,8,16,16,-1,1,1,1,2,2,2,2,4,2,4,2,8,2, 8,4,8,4,16,4,16,4,16,4,16,8,2,8,4,8,4,8,4,8,8,12,4,4],[1,1,2,3,4,20,8,24, 16,16,-1,1,1,1,2,2,2,2,4,2,4,2,8,2,8,4,8,4,16,4,16,4,16,4,16,8,4,8,4,8, 8,8,8,20,4,4,8,8],[1,1,2,19,4,28,8,16,-1,1,1,1,2,2,2,4,2,4,2,4,2,4,2,4, 4,8,4,8,4,8,4,8,8,8,8,4,4,4,4,4,4,4,4,4,4,4],[1,1,2,11,4,36,8,16,-1,1,1, 1,2,2,2,4,2,4,2,4,4,4,4,4,4,8,4,8,4,8,4,8,8,8,8,4,4,4,4,4,4,12,4,4,4,4], [1,1,2,19,4,44,-1,1,1,1,2,2,2,4,2,4,2,4,4,4,4,4,4,8,2,8,4,8,4,8,4,8,4,4, 4,4,4,4,4,4,4,4,4,4],[1,1,2,11,4,52,-1,1,1,1,2,2,2,4,2,4,2,4,4,4,4,4,4, 8,4,8,4,8,4,8,4,8,4,4,4,4,4,4,4,4,4,4,4,12],[1,1,2,11,4,20,8,32,-1,1,1, 1,2,2,2,4,4,4,4,4,4,4,4,4,4,8,2,8,8,8,8,8,8,8,8,4,4,12,4,4,4,4,4,4,4,4], [1,1,2,3,4,28,8,32,-1,1,1,1,2,2,2,4,4,4,4,4,4,4,4,4,4,8,4,8,8,8,8,8,8,8, 8,4,4,20,4,4,4,4,4,4,4,4],[1,1,2,23,4,8,8,32,-1,1,1,1,2,2,2,2,4,2,4,2,8, 2,8,4,2,4,4,4,8,4,8,4,8,8,2,8,2,8,8,8,8,8,8,8,8,8],[1,1,2,15,4,16,8,32, -1,1,1,1,2,2,2,2,4,2,4,2,8,2,8,4,2,4,4,4,8,4,8,4,8,8,2,8,4,8,8,8,8,16,8, 8,8,8],[1,1,2,7,4,24,8,32,-1,1,1,1,2,2,2,2,4,2,4,2,8,2,8,4,2,4,4,4,8,4, 8,4,8,8,4,8,4,8,8,8,8,24,8,8,8,8],[1,1,2,27,4,36,-1,1,1,1,2,2,2,2,2,2,2, 4,2,4,2,4,2,4,2,4,2,4,4,4,4,4,4,8,4,8,4,8,4,4,4,4,4,4,4,12],[1,1,2,19,4, 44,-1,1,1,1,2,2,2,2,2,2,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,4,4,8,4,8,4,8,4,4, 4,4,4,4,4,20],[1,1,2,27,4,20,8,16,-1,1,1,1,2,2,2,2,4,2,4,4,2,4,2,4,2,4, 2,4,4,4,4,4,4,4,4,8,2,8,8,8,8,4,4,12,4,4,4,4],[1,1,2,19,4,28,8,16,-1,1, 1,1,2,2,2,2,4,2,4,4,2,4,2,4,2,4,2,4,4,4,4,4,4,4,4,8,4,8,8,8,8,4,4,20,4, 4,4,4],[1,1,2,19,4,28,8,16,-1,1,1,1,2,2,2,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4, 4,4,4,4,4,8,2,8,8,8,8,4,4,20,4,4,4,4],[1,1,2,11,4,36,8,16,-1,1,1,1,2,2, 2,2,4,2,4,4,2,4,2,4,4,4,4,4,4,4,4,4,4,4,4,8,4,8,8,8,8,4,4,28,4,4,4,4],[1, 1,2,33,4,2,8,4,16,8,32,16,-1,1,1,1,2,2,4,2,8,2,8,2,16,2,16,2,16,2,16,2, 32,2,32,2,32,2,32,2,32,2,32,2,32,2,32,16,2,16,2,2,2,2],[1,1,2,17,4,18,8, 4,16,8,32,16,-1,1,1,1,2,2,4,2,8,2,8,2,16,2,16,2,16,2,16,2,32,2,32,2,32, 2,32,2,32,2,32,2,32,2,32,16,2,16,4,18,2,2],[1,1,2,1,4,34,8,4,16,8,32,16, -1,1,1,1,2,2,4,2,8,2,8,2,16,2,16,2,16,2,16,2,32,2,32,2,32,2,32,2,32,2,32, 2,32,2,32,16,4,16,4,34,2,2]]; ############################################################################# ## #F GroupIdOps.Group66( ) . . . . . . . . . . . . . . groups of order 66 ## GroupIdOps.Group66 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList66, x ); # and return return rec( catalogue := [ 66, nr ], names := GroupIdOps.GroupName66[nr], size := 66 ); end; GroupIdOps.GroupName66 := [ [ "66" ], [ "D22x3" ], [ "S3x11" ], [ "D66" ] ]; GroupIdOps.GroupList66 := [ [ 1, 1, 2, 1, 3, 2, 6, 2, 11, 10, 22, 10, 33, 20, 66, 20 ], [ 1, 1, 2, 11, 3, 2, 6, 22, 11, 10, 33, 20 ], [ 1, 1, 2, 3, 3, 2, 11, 10, 22, 30, 33, 20 ], [ 1, 1, 2, 33, 3, 2, 11, 10, 33, 20 ] ]; ############################################################################# ## #F GroupIdOps.Group70( ) . . . . . . . . . . . . . . groups of order 70 ## GroupIdOps.Group70 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList70, x ); # and return return rec( catalogue := [ 70, nr ], names := GroupIdOps.GroupName70[nr], size := 70 ); end; GroupIdOps.GroupName70 := [ [ "70" ], [ "D14x5" ], [ "D10x7" ], [ "D70" ] ]; GroupIdOps.GroupList70 := [ [ 1, 1, 2, 1, 5, 4, 7, 6, 10, 4, 14, 6, 35, 24, 70, 24 ], [ 1, 1, 2, 7, 5, 4, 7, 6, 10, 28, 35, 24 ], [ 1, 1, 2, 5, 5, 4, 7, 6, 14, 30, 35, 24 ], [ 1, 1, 2, 35, 5, 4, 7, 6, 35, 24 ] ]; ############################################################################# ## #F GroupIdOps.Group72( ) . . . . . . . . . . . . . . groups of order 72 ## GroupIdOps.Group72 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList72, x ); # problem pair 36/47 if nr = 36 and Length(DerivedSeries(G)) = 4 then nr := 47; fi; # and return return rec( catalogue := [ 72, nr ], names := GroupIdOps.GroupName72[nr], size := 72 ); end; GroupIdOps.GroupName72 := [ ["2^3x3^2"], ["2^3x9"], ["2x4x3^2"], ["2x4x9"], ["8x3^2"], ["72"], ["3^2xD8"], ["3^2xQ8"], ["9xD8"], ["9xQ8"], ["2x6xS3"], ["6x6.2"], ["12xS3"], ["3x12.2"], ["6xA4"], ["2x(2^2x3).3"], ["2^2x3^2:2"], ["2x(3^2x2).2"], ["4x3^2:2"], ["(4x3^2).2"], ["2^2xD18"], ["2x18.2"], ["4xD18"], ["36.2"], [], ["3xD24"], ["3xQ8+S3"], ["Sl(2,3)x3"], ["(Q8x3).3"], ["3xS4"], ["2xS3^2"], ["S3x6.2"], ["(6.2)Y(6.2)"], ["2x3^2:4"], ["(2x3^2).4"], [], ["S3+D24"], ["Q8+(3^2:2)"], [], ["D72"], ["Q8+D18"], [], [], [], [], ["S3xA4"], [], [], [], [] ]; GroupIdOps.GroupList72 := [ # 72.2 - 72.5 swapped [ 1, 1, 2, 7, 3, 8, 6, 56 ], [ 1, 1, 2, 7, 3, 2, 6, 14, 9, 6, 18, 42 ], # 72.4 [ 1, 1, 2, 3, 3, 8, 4, 4, 6, 24, 12, 32 ], # 72.2 [ 1, 1, 2, 3, 3, 2, 4, 4, 6, 6, 9, 6, 12, 8, 18, 18, 36, 24 ], # 72.5 [ 1, 1, 2, 1, 3, 8, 4, 2, 6, 8, 8, 4, 12, 16, 24, 32], # 72.3 [ 1, 1, 2, 1, 3, 2, 4, 2, 6, 2, 8, 4, 9, 6, 12, 4, 18, 6, 24, 8, 36, 12, 72, 24 ], [ 1, 1, 2, 5, 3, 8, 4, 2, 6, 40, 12, 16 ], [ 1, 1, 2, 1, 3, 8, 4, 6, 6, 8, 12, 48 ], [ 1, 1, 2, 5, 3, 2, 4, 2, 6, 10, 9, 6, 12, 4, 18, 30, 36, 12 ], [ 1, 1, 2, 1, 3, 2, 4, 6, 6, 2, 9, 6, 12, 12, 18, 6, 36, 36 ], [ 1, 1, 2, 15, 3, 8, 6, 48 ], [ 1, 1, 2, 3, 3, 8, 4, 12, 6, 24, 12, 24 ], [ 1, 1, 2, 7, 3, 8, 4, 8, 6, 20, 12, 28 ], [ 1, 1, 2, 1, 3, 8, 4, 2, 6, 8, 8, 12, 12, 16, 24, 24 ], [ 1, 1, 2, 7, 3, 26, 6, 38 ], [ 1, 1, 2, 7, 3, 2, 6, 14, 9, 24, 18, 24 ], [ 1, 1, 2, 39, 3, 8, 6, 24 ], [ 1, 1, 2, 3, 3, 8, 4, 36, 6, 24 ], [ 1, 1, 2, 19, 3, 8, 4, 20, 6, 8, 12, 16 ], [ 1, 1, 2, 1, 3, 8, 4, 2, 6, 8, 8, 36, 12, 16 ], [ 1, 1, 2, 39, 3, 2, 6, 6, 9, 6, 18, 18 ], [ 1, 1, 2, 3, 3, 2, 4, 36, 6, 6, 9, 6, 18, 18 ], [ 1, 1, 2, 19, 3, 2, 4, 20, 6, 2, 9, 6, 12, 4, 18, 6, 36, 12 ], [ 1, 1, 2, 1, 3, 2, 4, 2, 6, 2, 8, 36, 9, 6, 12, 4, 18, 6, 36, 12 ], [ 1, 1, 2, 9, 3, 8, 4, 6, 6, 36, 12, 12 ], [ 1, 1, 2, 13, 3, 8, 4, 2, 6, 32, 12, 16 ], [ 1, 1, 2, 1, 3, 8, 4, 14, 6, 8, 12, 40 ], [ 1, 1, 2, 1, 3, 26, 4, 6, 6, 26, 12, 12 ], [ 1, 1, 2, 1, 3, 2, 4, 6, 6, 2, 9, 24, 12, 12, 18, 24 ], [ 1, 1, 2, 9, 3, 26, 4, 6, 6, 18, 12, 12 ], [ 1, 1, 2, 31, 3, 8, 6, 32 ], [ 1, 1, 2, 7, 3, 8, 4, 24, 6, 20, 12, 12 ], [ 1, 1, 2, 19, 3, 8, 4, 12, 6, 8, 12, 24 ], [ 1, 1, 2, 19, 3, 8, 4, 36, 6, 8 ], [ 1, 1, 2, 1, 3, 8, 4, 18, 6, 8, 8, 36 ], [ 1, 1, 2, 21, 3, 8, 4, 18, 6, 24 ], # 72.36 as 72.47 [ 1, 1, 2, 37, 3, 8, 4, 2, 6, 8, 12, 16 ], [ 1, 1, 2, 1, 3, 8, 4, 38, 6, 8, 12, 16 ], [ 1, 1, 2, 21, 3, 2, 4, 18, 6, 6, 9, 6, 18, 18 ], [ 1, 1, 2, 37, 3, 2, 4, 2, 6, 2, 9, 6, 12, 4, 18, 6, 36, 12 ], [ 1, 1, 2, 1, 3, 2, 4, 38, 6, 2, 9, 6, 12, 4, 18, 6, 36, 12 ], [ 1, 1, 2, 13, 3, 8, 4, 18, 6, 32 ], [ 1, 1, 2, 25, 3, 8, 4, 6, 6, 20, 12, 12 ], [ 1, 1, 2, 1, 3, 8, 4, 30, 6, 8, 12, 24 ], [ 1, 1, 2, 9, 3, 8, 4, 18, 8, 36 ], [ 1, 1, 2, 15, 3, 26, 6, 30 ], [ 1, 1, 2, 21, 3, 8, 4, 18, 6, 24 ], # 72.47 as 72.36 [ 1, 1, 2, 9, 3, 8, 4, 54 ], [ 1, 1, 2, 21, 3, 26, 4, 18, 6, 6 ], [ 1, 1, 2, 21, 3, 2, 4, 18, 6, 6, 9, 24 ] ]; ############################################################################# ## #F GroupIdOps.Group75( ) . . . . . . . . . . . . . . groups of order 75 ## GroupIdOps.Group75 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList75, x ); # and return return rec( catalogue := [ 75, nr ], names := GroupIdOps.GroupName75[nr], size := 75 ); end; GroupIdOps.GroupName75 := [ [ "15x5" ], [ "75" ], [ "5^2:3" ] ]; GroupIdOps.GroupList75 := [ [ 1, 1, 3, 2, 5, 24, 15, 48 ], [ 1, 1, 3, 2, 5, 4, 15, 8, 25, 20, 75, 40 ], [ 1, 1, 3, 50, 5, 24 ] ]; ############################################################################# ## #F GroupIdOps.Group78( ) . . . . . . . . . . . . . . groups of order 78 ## GroupIdOps.Group78 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList78, x ); # and return return rec( catalogue := [ 78, nr ], names := GroupIdOps.GroupName78[nr], size := 78 ); end; GroupIdOps.GroupName78 := [ ["78"], ["3xD26"], ["2x13:3"], ["13xS3"], ["13:6"], ["D78"] ]; GroupIdOps.GroupList78 := [ [ 1, 1, 2, 1, 3, 2, 6, 2, 13, 12, 26, 12, 39, 24, 78, 24 ], [ 1, 1, 2, 13, 3, 2, 6, 26, 13, 12, 39, 24 ], [ 1, 1, 2, 1, 3, 26, 6, 26, 13, 12, 26, 12 ], [ 1, 1, 2, 3, 3, 2, 13, 12, 26, 36, 39, 24 ], [ 1, 1, 2, 13, 3, 26, 6, 26, 13, 12 ], [ 1, 1, 2, 39, 3, 2, 13, 12, 39, 24 ] ]; ############################################################################# ## #F GroupIdOps.Group80( ) . . . . . . . . . . . . . . groups of order 80 ## GroupIdOps.Group80 := function( G ) local z1, z2, z3, nr, tmp; # look up element orders in group list z1 := Collected( List( Elements(G), x -> Order(G,x) ) ); # abelian or nor abelian? if IsAbelian(G) then z2 := 1; else z2 := 0; fi; # length of conjugacy classes # List(ConjugacyClasses(G),x->[Size(x),Order(G,Representative(x))]) z3 := List( Orbits( G, Elements(G) ), x -> [Length(x),Order(G,x[1])] ); z3 := Collected(z3); Sort(z3); # look up the group pattern nr := Position( GroupIdOps.GroupList80, Flat([z1,z2,z3]) ); # problem pair 7/10, 80.7=[12], 80.10=[4,8] see 48.?? if nr = 7 then if GroupIdOps.Nr4Squares(G) = [4,8] then nr := 10; fi; fi; # problem triple 29/32/33, 80.29=[44], 80.32=[20,24], 80.33=[4,40] if nr = 29 then tmp := GroupIdOps.Nr4Squares(G); if tmp = [20,24] then nr := 32; elif tmp = [4,40] then nr := 33; fi; fi; # and return return rec( catalogue := [ 80, nr ], names := GroupIdOps.GroupName80[nr], size := 80 ); end; GroupIdOps.GroupName80 := [ ["2^4x5"], ["2^2x20"], ["8x10"], ["4^2x5"], ["80"], ["10xD8"], ["10xQ8"], ["5xD8Y4"], [], ["5x(2x4).2"], [], ["5xD16"], ["5xQD16"], ["5xQ16"], ["2^3xD10"], ["4xD20"], ["2^2x10.2"], ["4x10.2"], ["8xD10"], ["2x20.2"], ["40.2"], ["2^2x5:4"], ["4x5:4"], ["2x10.4"], ["4Y(10.4)"], ["20.4"], [], ["2xD40"], ["2xQ8+D10"], [], [], [], ["(10.2)+Q8"], ["D40Y4"], [], [], ["D8xD10"], ["Q8xD10"], ["(10.2)YD8"], ["(10.2)YQ8"], [], ["(5:4)+Q8"], [], [], [], [], [], [], ["D80"], [], [], ["2^4:5"] ]; GroupIdOps.GroupList80 := [ [1,1,2,15,5,4,10,60,1,1,1,1,1,2,15,1,5,4,1,10,60], [1,1,2,7,4,8,5,4,10,28,20,32,1,1,1,1,1,2,7,1,4,8,1,5,4,1,10,28,1,20,32], [1,1,2,3,4,4,5,4,8,8,10,12,20,16,40,32,1,1,1,1,1,2,3,1,4,4,1,5,4,1,8,8, 1,10,12,1,20,16,1,40,32], [1,1,2,3,4,12,5,4,10,12,20,48,1,1,1,1,1,2,3,1,4,12,1,5,4,1,10,12,1,20,48], [1,1,2,1,4,2,5,4,8,4,10,4,16,8,20,8,40,16,80,32,1,1,1,1,1,2,1,1,4,2,1,5, 4,1,8,4,1,10,4,1,16,8,1,20,8,1,40,16,1,80,32], [1,1,2,11,4,4,5,4,10,44,20,16,0,1,1,1,1,2,3,1,5,4,1,10,12,2,2,4,2,4,2,2, 10,16,2,20,8], [1,1,2,3,4,12,5,4,10,12,20,48,0,1,1,1,1,2,3,1,5,4,1,10,12,2,4,6,2,20,24], [1,1,2,7,4,8,5,4,10,28,20,32,0,1,1,1,1,2,1,1,4,2,1,5,4,1,10,4,1,20,8,2, 2,3,2,4,3,2,10,12,2,20,12], [1,1,2,7,4,8,5,4,10,28,20,32,0,1,1,1,1,2,3,1,5,4,1,10,12,2,2,2,2,4,4,2, 10,8,2,20,16], [1,1,2,3,4,12,5,4,10,12,20,48,0,1,1,1,1,2,3,1,5,4,1,10,12,2,4,6,2,20,24], [1,1,2,3,4,4,5,4,8,8,10,12,20,16,40,32,0,1,1,1,1,2,1,1,4,2,1,5,4,1,10,4, 1,20,8,2,2,1,2,4,1,2,8,4,2,10,4,2,20,4,2,40,16], [1,1,2,9,4,2,5,4,8,4,10,36,20,8,40,16,0,1,1,1,1,2,1,1,5,4,1,10,4,2,4,1,2, 8,2,2,20,4,2,40,8,4,2,2,4,10,8], [1,1,2,5,4,6,5,4,8,4,10,20,20,24,40,16,0,1,1,1,1,2,1,1,5,4,1,10,4,2,4,1, 2,8,2,2,20,4,2,40,8,4,2,1,4,4,1,4,10,4,4,20,4], [1,1,2,1,4,10,5,4,8,4,10,4,20,40,40,16,0,1,1,1,1,2,1,1,5,4,1,10,4,2,4,1, 2,8,2,2,20,4,2,40,8,4,4,2,4,20,8], [1,1,2,47,5,4,10,28,0,1,1,1,1,2,7,2,5,2,2,10,14,5,2,8], [1,1,2,23,4,24,5,4,10,12,20,16,0,1,1,1,1,2,3,1,4,4,2,5,2,2,10,6,2,20,8, 5,2,4,5,4,4], [1,1,2,7,4,40,5,4,10,28,0,1,1,1,1,2,7,2,5,2,2,10,14,5,4,8], [1,1,2,3,4,44,5,4,10,12,20,16,0,1,1,1,1,2,3,1,4,4,2,5,2,2,10,6,2,20,8,5, 4,8], [1,1,2,11,4,12,5,4,8,24,10,4,20,8,40,16,0,1,1,1,1,2,1,1,4,2,1,8,4,2,5,2, 2,10,2,2,20,4,2,40,8,5,2,2,5,4,2,5,8,4], [1,1,2,3,4,4,5,4,8,40,10,12,20,16,0,1,1,1,1,2,3,1,4,4,2,5,2,2,10,6,2,20, 8,5,8,8], [1,1,2,1,4,2,5,4,8,4,10,4,16,40,20,8,40,16,0,1,1,1,1,2,1,1,4,2,1,8,4,2, 5,2,2,10,2,2,20,4,2,40,8,5,16,8], [1,1,2,23,4,40,5,4,10,12,0,1,1,1,1,2,3,4,5,1,4,10,3,5,2,4,5,4,8], [1,1,2,11,4,52,5,4,10,4,20,8,0,1,1,1,1,2,1,1,4,2,4,5,1,4,10,1,4,20,2,5, 2,2,5,4,10], [1,1,2,3,4,20,5,4,8,40,10,12,0,1,1,1,1,2,3,4,5,1,4,10,3,5,4,4,5,8,8], [1,1,2,11,4,12,5,4,8,40,10,4,20,8,0,1,1,1,1,2,1,1,4,2,4,5,1,4,10,1,4,20, 2,5,2,2,5,4,2,5,8,8], [1,1,2,1,4,2,5,4,8,20,10,4,16,40,20,8,0,1,1,1,1,2,1,1,4,2,4,5,1,4,10,1, 4,20,2,5,8,4,5,16,8], [1,1,2,27,4,20,5,4,10,28,0,1,1,1,1,2,3,2,2,2,2,5,2,2,10,14,10,2,2,10,4,2], [1,1,2,43,4,4,5,4,10,12,20,16,0,1,1,1,1,2,3,2,4,2,2,5,2,2,10,6,2,20,8, 10,2,4], [1,1,2,3,4,44,5,4,10,12,20,16,0,1,1,1,1,2,3,2,4,2,2,5,2,2,10,6,2,20,8, 10,4,4], [1,1,2,7,4,40,5,4,10,28,0,1,1,1,1,2,3,2,2,2,2,5,2,2,10,14,10,4,4], [1,1,2,23,4,24,5,4,10,12,20,16,0,1,1,1,1,2,3,2,4,2,2,5,2,2,10,6,2,20,8, 10,2,2,10,4,2], [1,1,2,3,4,44,5,4,10,12,20,16,0,1,1,1,1,2,3,2,4,2,2,5,2,2,10,6,2,20,8, 10,4,4], [1,1,2,3,4,44,5,4,10,12,20,16,0,1,1,1,1,2,3,2,4,2,2,5,2,2,10,6,2,20,8, 10,4,4], [1,1,2,23,4,24,5,4,10,12,20,16,0,1,1,1,1,2,1,1,4,2,2,2,1,2,4,1,2,5,2,2, 10,6,2,20,8,10,2,2,10,4,2], [1,1,2,3,4,4,5,4,8,40,10,12,20,16,0,1,1,1,1,2,1,1,4,2,2,2,1,2,4,1,2,5,2, 2,10,6,2,20,8,10,8,4], [1,1,2,11,4,12,5,4,8,24,10,4,20,8,40,16,0,1,1,1,1,2,1,1,4,2,2,5,2,2,8,2, 2,10,2,2,20,4,2,40,8,10,2,1,10,4,1,10,8,2], [1,1,2,35,4,12,5,4,10,20,20,8,0,1,1,1,1,2,1,2,2,2,2,4,1,2,5,2,2,10,2,4, 10,4,4,20,2,5,2,2,10,2,2,10,4,1], [1,1,2,11,4,36,5,4,10,4,20,24,0,1,1,1,1,2,1,2,4,3,2,5,2,2,10,2,4,20,6,5, 2,2,10,4,3], [1,1,2,15,4,32,5,4,10,20,20,8,0,1,1,1,1,2,1,2,2,2,2,4,1,2,5,2,2,10,2,4, 10,4,4,20,2,5,4,2,10,2,1,10,4,2], [1,1,2,31,4,16,5,4,10,4,20,24,0,1,1,1,1,2,1,2,4,3,2,5,2,2,10,2,4,20,6,5, 4,2,10,2,3], [1,1,2,23,4,40,5,4,10,12,0,1,1,1,1,2,1,2,2,1,4,5,1,4,10,3,5,2,2,10,2,1, 10,4,4], [1,1,2,11,4,52,5,4,10,4,20,8,0,1,1,1,1,2,1,2,4,1,4,5,1,4,10,1,4,20,2,5, 2,2,10,4,5], [1,1,2,3,4,20,5,4,8,40,10,12,0,1,1,1,1,2,1,2,2,1,4,5,1,4,10,3,5,4,2,10, 4,1,10,8,4], [1,1,2,11,4,12,5,4,8,40,10,4,20,8,0,1,1,1,1,2,1,2,4,1,4,5,1,4,10,1,4,20, 2,5,4,2,10,2,1,10,8,4], [1,1,2,25,4,2,5,4,8,20,10,20,20,8,0,1,1,1,1,2,1,2,4,1,2,5,2,2,10,2,4,2, 1,4,10,4,4,20,2,10,8,2,20,2,1], [1,1,2,5,4,22,5,4,8,20,10,20,20,8,0,1,1,1,1,2,1,2,4,1,2,5,2,2,10,2,4,2, 1,4,10,4,4,20,2,10,8,2,20,4,1], [1,1,2,21,4,6,5,4,8,20,10,4,20,24,0,1,1,1,1,2,1,2,4,1,2,5,2,2,10,2,4,4, 1,4,20,6,10,8,2,20,2,1], [1,1,2,1,4,26,5,4,8,20,10,4,20,24,0,1,1,1,1,2,1,2,4,1,2,5,2,2,10,2,4,4, 1,4,20,6,10,8,2,20,4,1], [1,1,2,41,4,2,5,4,8,4,10,4,20,8,40,16,0,1,1,1,1,2,1,2,4,1,2,5,2,2,8,2,2, 10,2,2,20,4,2,40,8,20,2,2], [1,1,2,21,4,22,5,4,8,4,10,4,20,8,40,16,0,1,1,1,1,2,1,2,4,1,2,5,2,2,8,2, 2,10,2,2,20,4,2,40,8,20,2,1,20,4,1], [1,1,2,1,4,42,5,4,8,4,10,4,20,8,40,16,0,1,1,1,1,2,1,2,4,1,2,5,2,2,8,2,2, 10,2,2,20,4,2,40,8,20,4,2], [1,1,2,15,5,64,0,1,1,1,5,2,3,16,5,4]]; ############################################################################# ## #F GroupIdOps.Group81( ) . . . . . . . . . . . . . . groups of order 81 ## GroupIdOps.Group81 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList81, x ); # problem pair 1/6, 81.1 is abelian, 81.6 is not if nr = 1 and not IsAbelian(G) then nr := 6; fi; # problem pair 3/11, 81.3 is abelian, 81.11 is not if nr = 3 and not IsAbelian(G) then nr := 11; fi; # problem triple 4/9/15, check conjugacy classes if nr = 4 and not IsAbelian(G) then if Length(ConjugacyClasses(G))=33 then nr := 9; else nr := 15; fi; fi; # problem tuple 2/7/8/10/14, check conjugacy classes if nr = 2 and not IsAbelian(G) then if Length(NormalSubgroups(G)) = 32 then nr := 7; elif Length(NormalSubgroups(G)) = 14 then nr := 8; elif Length(NormalSubgroups(G)) = 29 then nr := 10; else nr := 14; fi; fi; # and return return rec( catalogue := [ 81, nr ], names := GroupIdOps.GroupName81[nr], size := 81 ); end; GroupIdOps.GroupName81 := [ ["3^4"], ["3^2x9"], ["27x3"], ["9^2"], ["81"], ["3x3^3:3"], ["3x9:3"], [], [], ["(3^2:3)Y9"], ["27:3"], ["3$3"], [], [], [] ]; GroupIdOps.GroupList81 := [ [ 1, 1, 3, 80 ], [ 1, 1, 3, 26, 9, 54 ], [ 1, 1, 3, 8, 9, 18, 27, 54 ], # 81.4 in catalogue [ 1, 1, 3, 8, 9, 72 ], # 81.3 in catalogue [ 1, 1, 3, 2, 9, 6, 27, 18, 81, 54 ], [ 1, 1, 3, 80 ], [ 1, 1, 3, 26, 9, 54 ], [ 1, 1, 3, 26, 9, 54 ], [ 1, 1, 3, 8, 9, 72 ], [ 1, 1, 3, 26, 9, 54 ], [ 1, 1, 3, 8, 9, 18, 27, 54 ], [ 1, 1, 3, 44, 9, 36 ], [ 1, 1, 3, 62, 9, 18 ], [ 1, 1, 3, 26, 9, 54 ], [ 1, 1, 3, 8, 9, 72 ] ]; ############################################################################# ## #F GroupIdOps.Group84( ) . . . . . . . . . . . . . . groups of order 84 ## GroupIdOps.Group84 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList84, x ); # and return return rec( catalogue := [ 84, nr ], names := GroupIdOps.GroupName84[nr], size := 84 ); end; GroupIdOps.GroupName84 := [ ["2^2x21"], ["84"], ["14xS3"], ["7x6.2"], ["A4x7"], ["D14x6"], ["14.2x3"], ["2^2x7:3"], ["4x7:3"], ["2x7:6"], ["(14.2):3"], ["D84"], ["42.2"], ["S3xD14"], ["A4+(7:3)"] ]; GroupIdOps.GroupList84 := [ [ 1, 1, 2, 3, 3, 2, 6, 6, 7, 6, 14, 18, 21, 12, 42, 36 ], [ 1, 1, 2, 1, 3, 2, 4, 2, 6, 2, 7, 6, 12, 4, 14, 6, 21, 12, 28, 12, 42, 12,84, 24 ], [ 1, 1, 2, 7, 3, 2, 6, 2, 7, 6, 14, 42, 21, 12, 42, 12 ], [ 1, 1, 2, 1, 3, 2, 4, 6, 6, 2, 7, 6, 14, 6, 21, 12, 28, 36, 42, 12 ], [ 1, 1, 2, 3, 3, 8, 7, 6, 14, 18, 21, 48 ], [ 1, 1, 2, 15, 3, 2, 6, 30, 7, 6, 14, 6, 21, 12, 42, 12 ], [ 1, 1, 2, 1, 3, 2, 4, 14, 6, 2, 7, 6, 12, 28, 14, 6, 21, 12, 42, 12 ], [ 1, 1, 2, 3, 3, 14, 6, 42, 7, 6, 14, 18 ], [ 1, 1, 2, 1, 3, 14, 4, 2, 6, 14, 7, 6, 12, 28, 14, 6, 28, 12 ], [ 1, 1, 2, 15, 3, 14, 6, 42, 7, 6, 14, 6 ], [ 1, 1, 2, 1, 3, 14, 4, 14, 6, 14, 7, 6, 12, 28, 14, 6 ], [ 1, 1, 2, 43, 3, 2, 6, 2, 7, 6, 14, 6, 21, 12, 42, 12 ], [ 1, 1, 2, 1, 3, 2, 4, 42, 6, 2, 7, 6, 14, 6, 21, 12, 42, 12 ], [ 1, 1, 2, 31, 3, 2, 6, 14, 7, 6, 14, 18, 21, 12 ], [ 1, 1, 2, 3, 3, 56, 7, 6, 14, 18 ] ]; ############################################################################# ## #F GroupIdOps.Group88( ) . . . . . . . . . . . . . . groups of order 88 ## GroupIdOps.Group88 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList88, x ); # and return return rec( catalogue := [ 88, nr ], names := GroupIdOps.GroupName88[nr], size := 88 ); end; GroupIdOps.GroupName88 := [ ["2^3x11"], ["2x44"], ["88"], ["11xD8"], ["11xQ8"], ["2xD44"], ["4xD22"], ["2x22.2"], ["44.2"], [], ["D88"], ["Q8+D22"] ]; GroupIdOps.GroupList88 := [ [ 1, 1, 2, 7, 11, 10, 22, 70 ], [ 1, 1, 2, 3, 4, 4, 11, 10, 22, 30, 44, 40 ], [ 1, 1, 2, 1, 4, 2, 8, 4, 11, 10, 22, 10, 44, 20, 88, 40 ], [ 1, 1, 2, 5, 4, 2, 11, 10, 22, 50, 44, 20 ], [ 1, 1, 2, 1, 4, 6, 11, 10, 22, 10, 44, 60 ], [ 1, 1, 2, 47, 11, 10, 22, 30 ], [ 1, 1, 2, 23, 4, 24, 11, 10, 22, 10, 44, 20 ], [ 1, 1, 2, 3, 4, 44, 11, 10, 22, 30 ], [ 1, 1, 2, 1, 4, 2, 8, 44, 11, 10, 22, 10, 44, 20 ], [ 1, 1, 2, 25, 4, 22, 11, 10, 22, 30 ], [ 1, 1, 2, 45, 4, 2, 11, 10, 22, 10, 44, 20 ], [ 1, 1, 2, 1, 4, 46, 11, 10, 22, 10, 44, 20 ] ]; ############################################################################# ## #F GroupIdOps.Group90( ) . . . . . . . . . . . . . . groups of order 90 ## GroupIdOps.Group90 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList90, x ); # and return return rec( catalogue := [ 90, nr ], names := GroupIdOps.GroupName90[nr], size := 90 ); end; GroupIdOps.GroupName90 := [ ["3x30"], ["90"], ["15xS3"], ["5x3^2:2"], ["5xD18"], ["3^2xD10"], ["9xD10"], ["3xD30"], ["(3^2x5):2"], ["D90"] ]; GroupIdOps.GroupList90 := [ [ 1, 1, 2, 1, 3, 8, 5, 4, 6, 8, 10, 4, 15, 32, 30, 32 ], [ 1, 1, 2, 1, 3, 2, 5, 4, 6, 2, 9, 6, 10, 4, 15, 8, 18, 6, 30, 8, 45, 24, 90, 24 ], [ 1, 1, 2, 3, 3, 8, 5, 4, 6, 6, 10, 12, 15, 32, 30, 24 ], [ 1, 1, 2, 9, 3, 8, 5, 4, 10, 36, 15, 32 ], [ 1, 1, 2, 9, 3, 2, 5, 4, 9, 6, 10, 36, 15, 8, 45, 24 ], [ 1, 1, 2, 5, 3, 8, 5, 4, 6, 40, 15, 32 ], [ 1, 1, 2, 5, 3, 2, 5, 4, 6, 10, 9, 6, 15, 8, 18, 30, 45, 24 ], [ 1, 1, 2, 15, 3, 8, 5, 4, 6, 30, 15, 32 ], [ 1, 1, 2, 45, 3, 8, 5, 4, 15, 32 ], [ 1, 1, 2, 45, 3, 2, 5, 4, 9, 6, 15, 8, 45, 24 ] ]; ############################################################################# ## #F GroupIdOps.Group96( ) . . . . . . . . . . . . . . groups of order 96 ## GroupIdOps.Group96 := function( G ) local z1, z3, nr, z2; # look up element orders in group list z1 := Collected( List( Elements(G), x -> Order(G,x) ) ); # abelian or not abelian? if IsAbelian(G) then z2 := 1; else z2 := 0; fi; # length of conjugacy classes # List(ConjugacyClasses(G),x->[Size(x),Order(G,Representative(x))]) z3 := List( Orbits( G, Elements(G) ), x -> [Length(x),Order(G,x[1])] ); z3 := Collected(z3); Sort(z3); # loop up group pattern nr := Position( GroupIdOps.GroupList96, Flat([z1,z2,z3]) ); # for problem cases use squares if nr in [30,31,32,35,50,51,53,62,78,79,80,94,95,97,98,99,101, 102,139,140,150,166,215] then z1 := GroupIdOps.GroupSquares96[nr]; # squares of elements of order 4 or 8 if z1[1] = 4 then z2 := GroupIdOps.Nr4Squares(G); else z2 := GroupIdOps.Nr8Squares(G); fi; z3 := Position( z1[3], z2 ); nr := z1[2][z3]; fi; # and return return rec( catalogue := [ 96, nr ], names := GroupIdOps.GroupName96[nr], size := 96 ); end; GroupIdOps.GroupName96 := [ ["2^5x3"], ["2^3x12"], ["2^2x24"], ["6x4^2"], ["2x48"], ["4x24"], ["96"], ["#96.01a"], ["#96.01b"], ["#96.01c"], ["#96.02a"], ["#96.02b"], ["#96.02c"], ["#96.03a"], ["#96.03b"], ["#96.03c"], ["#96.04a"], ["#96.04b"], ["#96.04c"], ["#96.05a"], ["#96.05b"], ["#96.06a"], ["#96.06b"], ["#96.07a"], ["#96.008"], ["#96.08a"], ["#96.08b"], ["#96.08c"], ["#96.08d"], ["#96.009"], ["#96.09a"], ["#96.09b"], ["#96.09c"], ["#96.09d"], ["#96.09e"], ["#96.010"], ["#96.10a"], ["#96.10b"], ["#96.10c"], ["#96.10d"], ["#96.10e"], ["#96.011"], ["#96.11a"], ["#96.11b"], ["#96.11c"], ["#96.012"], ["#96.12a"], ["#96.12b"], ["#96.12c"], ["#96.013"], ["#96.13a"], ["#96.13b"], ["#96.13c"], ["#96.014"], ["#96.14a"], ["#96.14b"], ["#96.14c"], ["#96.14d"], ["#96.14e"], ["#96.015"], ["#96.15a"], ["#96.15b"], ["#96.15c"], ["#96.15d"], ["#96.016"], ["#96.16a"], ["#96.16b"], ["#96.16c"], ["#96.16d"], ["#96.017"], ["#96.17a"], ["#96.17b"], ["#96.17c"], ["#96.17d"], ["#96.018"], ["#96.18a"], ["#96.18b"], ["#96.019"], ["#96.19a"], ["#96.19b"], ["#96.020"], ["#96.20a"], ["#96.20b"], ["#96.021"], ["#96.21a"], ["#96.21b"], ["#96.022"], ["#96.22a"], ["#96.22b"], ["#96.023"], ["#96.23a"], ["#96.23b"], ["#96.23c"], ["#96.024"], ["#96.24a"], ["#96.24b"], ["#96.24c"], ["#96.24d"], ["#96.025"], ["#96.25a"], ["#96.25b"], ["#96.25c"], ["#96.026"], ["#96.26a"], ["#96.26b"], ["#96.26c"], ["#96.26d"], ["#96.26e"], ["#96.027"], ["#96.27a"], ["#96.27b"], ["#96.27c"], ["#96.028"], ["#96.28a"], ["#96.28b"], ["#96.28c"], ["#96.029"], ["#96.29a"], ["#96.29b"], ["#96.030"], ["#96.30a"], ["#96.30b"], ["#96.031"], ["#96.31a"], ["#96.31b"], ["#96.31c"], ["#96.032"], ["#96.32a"], ["#96.32b"], ["#96.033"], ["#96.33a"], ["#96.33b"], ["#96.33c"], ["#96.33d"], ["#96.034"], ["#96.34a"], ["#96.34b"], ["#96.34c"], ["#96.035"], ["#96.35b"], ["#96.35c"], ["#96.036"], ["#96.36a"], ["#96.36b"], ["#96.36c"], ["#96.36d"], ["#96.36e"], ["#96.037"], ["#96.37a"], ["#96.37b"], ["#96.37c"], ["#96.37d"], ["#96.37e"], ["#96.038"], ["#96.38a"], ["#96.38b"], ["#96.38c"], ["#96.38d"], ["#96.38e"], ["#96.039"], ["#96.39a"], ["#96.39b"], ["#96.39c"], ["#96.39d"], ["#96.040"], ["#96.40a"], ["#96.40b"], ["#96.40c"], ["#96.041"], ["#96.41a"], ["#96.41b"], ["#96.41c"], ["#96.41d"], ["#96.042"], ["#96.42a"], ["#96.42b"], ["#96.42c"], ["#96.42d"], ["#96.043"], ["#96.43a"], ["#96.43b"], ["#96.43c"], ["#96.044"], ["#96.44a"], ["#96.44b"], ["#96.44c"], ["#96.44d"], ["#96.44e"], ["#96.045"], ["#96.45a"], ["#96.45b"], ["#96.45c"], ["#96.45d"], ["#96.45e"], ["#96.046"], ["#96.46a"], ["#96.46b"], ["#96.047"], ["#96.47a"], ["#96.47b"], ["#96.048"], ["#96.48a"], ["#96.48b"], ["#96.049"], ["#96.49a"], ["#96.49b"], ["#96.050"], ["#96.50a"], ["#96.50b"], ["#96.50c"], ["#96.051"], ["#96.51a"], ["#96.51b"], ["#96.052"], ["#96.053"], ["#96.054"], ["#96.055"], ["#96.056"], ["#96.057"], ["#96.058"], ["#96.059"], ["#96.060"], ["#96.061"], ["#96.062"], ["#96.063"], ["#96.064"], ["#96.065"], ["#96.066"], ["#96.067"], ["#96.068"], ["#96.35a"] ]; GroupIdOps.GroupList96 := [ [1,1,2,31,3,2,6,62,1,1,1,1,1,2,31,1,3,2,1,6,62],[1,1,2,15,3,2,4,16, 6,30,12,32,1,1,1,1,1,2,15,1,3,2,1,4,16,1,6,30,1,12,32],[1,1,2,7,3,2,4,8,6,14, 8,16,12,16,24,32,1,1,1,1,1,2,7,1,3,2,1,4,8,1,6,14,1,8,16,1,12,16,1,24,32],[1, 1,2,7,3,2,4,24,6,14,12,48,1,1,1,1,1,2,7,1,3,2,1,4,24,1,6,14,1,12,48],[1,1,2, 3,3,2,4,4,6,6,8,8,12,8,16,16,24,16,48,32,1,1,1,1,1,2,3,1,3,2,1,4,4,1,6,6,1,8, 8,1,12,8,1,16,16,1,24,16,1,48,32],[1,1,2,3,3,2,4,12,6,6,8,16,12,24,24,32,1,1, 1,1,1,2,3,1,3,2,1,4,12,1,6,6,1,8,16,1,12,24,1,24,32],[1,1,2,1,3,2,4,2,6,2,8, 4,12,4,16,8,24,8,32,16,48,16,96,32,1,1,1,1,1,2,1,1,3,2,1,4,2,1,6,2,1,8,4,1, 12,4,1,16,8,1,24,8,1,32,16,1,48,16,1,96,32],[1,1,2,63,3,2,6,30,0,1,1,1,1,2, 15,2,3,1,2,6,15,3,2,16],[1,1,2,31,3,8,6,56,0,1,1,1,1,2,7,3,2,8,4,3,2,4,6,14], [1,1,2,31,3,32,6,32,0,1,1,1,1,2,1,3,2,10,16,3,2,16,6,2],[1,1,2,31,3,2,4,32, 6,14,12,16,0,1,1,1,1,2,7,1,4,8,2,3,1,2,6,7,2,12,8,3,2,8,3,4,8],[1,1,2,15,3, 2,4,48,6,30,0,1,1,1,1,2,15,2,3,1,2,6,15,3,4,16],[1,1,2,15,3,8,4,16,6,24,12, 32,0,1,1,1,1,2,3,1,4,4,3,2,4,3,4,4,4,3,2,4,6,6,4,12,8],[1,1,2,15,3,2,4,48,6, 6,12,24,0,1,1,1,1,2,3,1,4,12,2,3,1,2,6,3,2,12,12,3,2,4,3,4,12],[1,1,2,7,3,2, 4,56,6,14,12,16,0,1,1,1,1,2,7,1,4,8,2,3,1,2,6,7,2,12,8,3,4,16],[1,1,2,7,3,32, 4,24,6,32,0,1,1,1,1,2,1,3,2,2,3,4,8,16,3,2,16,6,2],[1,1,2,15,3,2,4,16,6,6,8, 32,12,8,24,16,0,1,1,1,1,2,3,1,4,4,1,8,8,2,3,1,2,6,3,2,12,4,2,24,8,3,2,4,3,4, 4,3,8,8],[1,1,2,7,3,2,4,8,6,14,8,48,12,16,0,1,1,1,1,2,7,1,4,8,2,3,1,2,6,7,2, 12,8,3,8,16],[1,1,2,7,3,8,4,8,6,8,8,16,12,16,24,32,0,1,1,1,1,2,1,1,4,2,1,8, 4,3,2,2,3,4,2,3,8,4,4,3,2,4,6,2,4,12,4,4,24,8],[1,1,2,3,3,2,4,28,6,6,8,32,12, 8,24,16,0,1,1,1,1,2,3,1,4,4,1,8,8,2,3,1,2,6,3,2,12,4,2,24,8,3,4,8,3,8,8],[1, 1,2,3,3,2,4,12,6,6,8,48,12,24,0,1,1,1,1,2,3,1,4,12,2,3,1,2,6,3,2,12,12,3,8, 16],[1,1,2,7,3,2,4,8,6,2,8,16,12,4,16,32,24,8,48,16,0,1,1,1,1,2,1,1,4,2,1,8, 4,1,16,8,2,3,1,2,6,1,2,12,2,2,24,4,2,48,8,3,2,2,3,4,2,3,8,4,3,16,8],[1,1,2,3, 3,2,4,4,6,6,8,8,12,8,16,48,24,16,0,1,1,1,1,2,3,1,4,4,1,8,8,2,3,1,2,6,3,2,12, 4,2,24,8,3,16,16],[1,1,2,1,3,2,4,2,6,2,8,4,12,4,16,8,24,8,32,48,48,16,0, 1,1,1,1,2,1,1,4,2,1,8,4,1,16,8,2,3,1,2,6,1,2,12,2,2,24,4,2,48,8,3,32,16], [1,1,2,23,3,2,4,8,6,46,12,16,0,1,1,1,1,2,7,1,3,2,1,6,14,2,2,8,2,4,4,2,6, 16,2,12,8],[1,1,2,55,3,2,4,8,6,14,12,16,0,1,1,1,1,2,7,2,3,1,2,4,4,2,6,7, 2,12,8,6,2,8],[1,1,2,47,3,2,4,16,6,22,12,8,0,1,1,1,1,2,3,2,2,4,2,3,1,2, 4,2,2,6,3,3,2,4,4,6,4,4,12,2,6,2,4,6,4,2],[1,1,2,39,3,2,4,24,6,30,0,1,1, 1,1,2,7,2,2,4,2,3,1,2,6,15,6,2,4,6,4,4],[1,1,2,23,3,8,4,8,6,40,12,16,0, 1,1,1,1,2,1,2,2,2,2,4,1,3,2,2,4,3,2,4,6,2,6,2,2,6,4,1,8,6,4,8,12,2],[1, 1,2,7,3,2,4,24,6,14,12,48,0,1,1,1,1,2,7,1,3,2,1,6,14,2,4,12,2,12,24],[1, 1,2,7,3,2,4,56,6,14,12,16,0,1,1,1,1,2,7,2,3,1,2,4,4,2,6,7,2,12,8,6,4,8], [1,1,2,15,3,2,4,48,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,3,2,4,4,12, 6,6,4,6],[1,1,2,7,3,8,4,24,6,8,12,48,0,1,1,1,1,2,1,2,4,3,3,2,2,4,3,2,4, 6,2,6,4,3,8,12,6],[1,1,2,7,3,8,4,24,6,56,0,1,1,1,1,2,7,4,3,2,4,6,14,6,4, 4],[1,1,2,7,3,32,4,24,6,32,0,1,1,1,1,2,1,3,2,2,6,4,4,16,3,2,16,6,2],[1, 1,2,15,3,2,4,16,6,30,12,32,0,1,1,1,1,2,3,1,3,2,1,4,4,1,6,6,1,12,8,2,2,6, 2,4,6,2,6,12,2,12,12],[1,1,2,31,3,2,4,32,6,14,12,16,0,1,1,1,1,2,3,1,4,4, 2,2,2,2,3,1,2,4,2,2,6,7,2,12,8,6,2,4,6,4,4],[1,1,2,31,3,2,4,32,6,14,12, 16,0,1,1,1,1,2,1,1,4,2,2,2,3,2,3,1,2,4,3,2,6,1,2,12,2,3,2,2,3,4,2,4,6,3, 4,12,3,6,2,3,6,4,3],[1,1,2,23,3,2,4,40,6,22,12,8,0,1,1,1,1,2,3,2,2,4,2, 3,1,2,4,2,2,6,3,3,4,4,4,6,4,4,12,2,6,2,2,6,4,4],[1,1,2,39,3,2,4,24,6,6, 12,24,0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,3,4,4,4,12,6,6,2,6],[1,1,2,15,3,8, 4,16,6,24,12,32,0,1,1,1,1,2,3,1,4,4,4,3,2,4,6,6,4,12,8,6,2,2,6,4,2],[1, 1,2,15,3,2,4,16,6,30,12,32,0,1,1,1,1,2,7,1,3,2,1,6,14,2,2,4,2,4,8,2,6,8, 2,12,16],[1,1,2,31,3,2,4,32,6,14,12,16,0,1,1,1,1,2,7,2,3,1,2,4,4,2,6,7, 2,12,8,6,2,4,6,4,4],[1,1,2,31,3,2,4,32,6,14,12,16,0,1,1,1,1,2,3,2,2,2,2, 3,1,2,4,4,2,6,3,3,2,4,4,6,2,4,12,4,6,2,2,6,4,4],[1,1,2,15,3,2,4,48,6,30, 0,1,1,1,1,2,7,2,2,4,2,3,1,2,6,15,6,4,8],[1,1,2,7,3,2,4,24,6,14,12,48,0, 1,1,1,1,2,7,1,3,2,1,6,14,2,4,12,2,12,24],[1,1,2,7,3,2,4,56,6,14,12,16,0, 1,1,1,1,2,7,2,3,1,2,4,4,2,6,7,2,12,8,6,4,8],[1,1,2,15,3,2,4,48,6,6,12,24, 0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,3,2,4,4,12,6,6,4,6],[1,1,2,7,3,2,4,56,6, 14,12,16,0,1,1,1,1,2,7,2,3,1,2,4,4,2,6,7,2,12,8,6,4,8],[1,1,2,7,3,2,4,8, 6,14,8,16,12,16,24,32,0,1,1,1,1,2,3,1,3,2,1,4,4,1,6,6,1,12,8,2,2,2,2,4, 2,2,6,4,2,8,8,2,12,4,2,24,16],[1,1,2,15,3,2,4,16,6,6,8,32,12,8,24,16,0, 1,1,1,1,2,3,1,4,4,2,3,1,2,6,3,2,8,4,2,12,4,2,24,8,6,2,2,6,4,2,6,8,4],[1, 1,2,15,3,2,4,16,6,6,8,32,12,8,24,16,0,1,1,1,1,2,1,1,4,2,2,2,1,2,3,1,2,4, 1,2,6,1,2,8,4,2,12,2,3,2,2,3,4,2,4,6,1,4,12,1,4,24,4,6,2,1,6,4,1,6,8,4], [1,1,2,7,3,2,4,8,6,14,8,48,12,16,0,1,1,1,1,2,3,1,4,4,2,2,2,2,3,1,2,4,2, 2,6,7,2,12,8,6,8,8],[1,1,2,11,3,2,4,20,6,22,12,40,0,1,1,1,1,2,3,1,3,2,1, 4,4,1,6,6,1,12,8,2,2,4,2,4,8,2,6,8,2,12,16],[1,1,2,19,3,2,4,44,6,14,12, 16,0,1,1,1,1,2,3,1,4,4,2,2,2,2,3,1,2,4,2,2,6,7,2,12,8,6,2,2,6,4,6],[1,1, 2,27,3,2,4,36,6,6,12,24,0,1,1,1,1,2,3,1,4,4,2,3,1,2,4,4,2,6,3,2,12,12,6, 2,4,6,4,4],[1,1,2,11,3,2,4,52,6,22,12,8,0,1,1,1,1,2,3,2,2,4,2,3,1,2,4,2, 2,6,3,3,4,4,4,6,4,4,12,2,6,4,6],[1,1,2,27,3,2,4,36,6,6,12,24,0,1,1,1,1, 2,3,2,3,1,2,4,6,2,6,3,3,4,4,4,12,6,6,2,4,6,4,2],[1,1,2,19,3,2,4,44,6,14, 12,16,0,1,1,1,1,2,3,2,2,2,2,3,1,2,4,4,2,6,3,3,4,4,4,6,2,4,12,4,6,2,2,6, 4,4],[1,1,2,3,3,2,4,28,6,6,12,56,0,1,1,1,1,2,3,1,3,2,1,4,4,1,6,6,1,12,8, 2,4,12,2,12,24],[1,1,2,3,3,2,4,60,6,6,12,24,0,1,1,1,1,2,3,1,4,4,2,3,1,2, 4,4,2,6,3,2,12,12,6,4,8],[1,1,2,3,3,2,4,60,6,6,12,24,0,1,1,1,1,2,3,2,3, 1,2,4,6,2,6,3,3,4,4,4,12,6,6,4,6],[1,1,2,3,3,2,4,60,6,6,12,24,0,1,1,1,1, 2,3,2,3,1,2,4,6,2,6,3,3,4,4,4,12,6,6,4,6],[1,1,2,3,3,8,4,28,6,24,12,32, 0,1,1,1,1,2,3,1,4,4,4,3,2,4,6,6,4,12,8,6,4,4],[1,1,2,7,3,2,4,24,6,14,12, 48,0,1,1,1,1,2,3,1,3,2,1,4,4,1,6,6,1,12,8,2,2,2,2,4,10,2,6,4,2,12,20],[1, 1,2,7,3,2,4,56,6,14,12,16,0,1,1,1,1,2,3,1,4,4,2,2,2,2,3,1,2,4,2,2,6,7,2, 12,8,6,4,8],[1,1,2,15,3,2,4,48,6,6,12,24,0,1,1,1,1,2,3,1,4,4,2,3,1,2,4, 4,2,6,3,2,12,12,6,2,2,6,4,6],[1,1,2,7,3,2,4,56,6,14,12,16,0,1,1,1,1,2,3, 2,2,2,2,3,1,2,4,4,2,6,3,3,4,4,4,6,2,4,12,4,6,4,6],[1,1,2,15,3,2,4,48,6, 6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,3,4,4,4,12,6,6,2,2,6,4,4],[1,1, 2,7,3,2,4,8,6,14,8,16,12,16,24,32,0,1,1,1,1,2,1,1,3,2,1,4,2,1,6,2,1,8,4, 1,12,4,1,24,8,2,2,3,2,4,3,2,6,6,2,8,6,2,12,6,2,24,12],[1,1,2,15,3,2,4,16, 6,6,8,32,12,8,24,16,0,1,1,1,1,2,1,1,4,2,1,8,4,2,2,1,2,3,1,2,4,1,2,6,3,2, 8,2,2,12,4,2,24,8,6,2,2,6,4,2,6,8,4],[1,1,2,15,3,2,4,16,6,6,8,32,12,8,24, 16,0,1,1,1,1,2,1,1,4,2,2,2,1,2,3,1,2,4,1,2,6,1,2,8,4,2,12,2,3,8,4,4,6,1, 4,12,1,4,24,4,6,2,2,6,4,2,6,8,2],[1,1,2,7,3,2,4,8,6,14,8,48,12,16,0,1,1, 1,1,2,1,1,4,2,2,2,3,2,3,1,2,4,3,2,6,1,2,12,2,3,8,4,4,6,3,4,12,3,6,8,6], [1,1,2,7,3,8,4,8,6,8,8,16,12,16,24,32,0,1,1,1,1,2,1,1,4,2,1,8,4,4,3,2,4, 6,2,4,12,4,4,24,8,6,2,1,6,4,1,6,8,2],[1,1,2,7,3,2,4,24,6,14,12,48,0,1,1, 1,1,2,7,1,3,2,1,6,14,2,4,12,2,12,24],[1,1,2,7,3,2,4,56,6,14,12,16,0,1,1, 1,1,2,7,2,3,1,2,4,4,2,6,7,2,12,8,6,4,8],[1,1,2,7,3,32,4,24,6,32,0,1,1,1, 1,2,1,3,2,2,6,4,4,16,3,2,16,6,2],[1,1,2,3,3,2,4,12,6,6,8,16,12,24,24,32, 0,1,1,1,1,2,3,1,3,2,1,4,4,1,6,6,1,12,8,2,4,4,2,8,8,2,12,8,2,24,16],[1,1, 2,3,3,2,4,28,6,6,8,32,12,8,24,16,0,1,1,1,1,2,3,1,4,4,2,3,1,2,6,3,2,8,4, 2,12,4,2,24,8,6,4,4,6,8,4],[1,1,2,3,3,2,4,12,6,6,8,48,12,24,0,1,1,1,1,2, 3,1,4,4,2,3,1,2,4,4,2,6,3,2,12,12,6,8,8],[1,1,2,7,3,2,4,8,6,14,8,16,12, 16,24,32,0,1,1,1,1,2,3,1,3,2,1,4,4,1,6,6,1,12,8,2,2,2,2,4,2,2,6,4,2,8,8, 2,12,4,2,24,16],[1,1,2,15,3,2,4,16,6,6,8,32,12,8,24,16,0,1,1,1,1,2,3,1, 4,4,2,3,1,2,6,3,2,8,4,2,12,4,2,24,8,6,2,2,6,4,2,6,8,4],[1,1,2,7,3,2,4,8, 6,14,8,48,12,16,0,1,1,1,1,2,3,1,4,4,2,2,2,2,3,1,2,4,2,2,6,7,2,12,8,6,8, 8],[1,1,2,3,3,2,4,12,6,6,8,16,12,24,24,32,0,1,1,1,1,2,3,1,3,2,1,4,4,1,6, 6,1,12,8,2,4,4,2,8,8,2,12,8,2,24,16],[1,1,2,3,3,2,4,12,6,6,8,48,12,24,0, 1,1,1,1,2,3,1,4,4,2,3,1,2,4,4,2,6,3,2,12,12,6,8,8],[1,1,2,3,3,2,4,28,6, 6,8,32,12,8,24,16,0,1,1,1,1,2,3,1,4,4,2,3,1,2,6,3,2,8,4,2,12,4,2,24,8,6, 4,4,6,8,4],[1,1,2,3,3,2,4,4,6,6,8,8,12,8,16,16,24,16,48,32,0,1,1,1,1,2, 1,1,3,2,1,4,2,1,6,2,1,8,4,1,12,4,1,24,8,2,2,1,2,4,1,2,6,2,2,8,2,2,12,2, 2,16,8,2,24,4,2,48,16],[1,1,2,7,3,2,4,8,6,2,8,16,12,4,16,32,24,8,48,16, 0,1,1,1,1,2,1,1,4,2,1,8,4,2,3,1,2,6,1,2,12,2,2,16,4,2,24,4,2,48,8,6,2,1, 6,4,1,6,8,2,6,16,4],[1,1,2,3,3,2,4,4,6,6,8,8,12,8,16,48,24,16,0,1,1,1,1, 2,1,1,4,2,1,8,4,2,2,1,2,3,1,2,4,1,2,6,3,2,8,2,2,12,4,2,24,8,6,16,8],[1, 1,2,19,3,2,4,4,6,38,8,8,12,8,24,16,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,2,2,8, 4,2,12,4,2,24,8,4,2,4,4,6,8],[1,1,2,51,3,2,4,4,6,6,8,8,12,8,24,16,0,1,1, 1,1,2,3,2,3,1,2,4,2,2,6,3,2,8,4,2,12,4,2,24,8,12,2,4],[1,1,2,39,3,2,4,8, 6,18,8,16,12,4,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8,2,3,2,2,4,2,2,4, 12,1,4,24,2,6,4,1,6,8,2,8,6,2,12,2,2],[1,1,2,35,3,2,4,4,6,22,8,24,12,8, 0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,2,2,4,6,4,4,12,2,6,8,4,12,2,2],[1,1,2, 11,3,2,4,12,6,22,8,8,12,24,24,16,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,2,2,8,4, 2,12,4,2,24,8,4,2,2,4,4,2,4,6,4,4,12,4],[1,1,2,27,3,2,4,28,6,6,8,8,12,8, 24,16,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,8,4,2,12,4,2,24,8,12,2,2,12,4,2], [1,1,2,23,3,2,4,24,6,10,8,16,12,12,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1, 2,8,2,3,2,2,4,2,1,4,4,1,4,12,1,4,24,2,6,4,1,6,8,2,8,6,1,8,12,1,12,2,1,12, 4,1],[1,1,2,11,3,2,4,28,6,22,8,24,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3, 4,2,2,4,6,4,4,12,2,6,8,4,12,4,2],[1,1,2,27,3,2,4,12,6,6,8,24,12,24,0,1, 1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,8,4,12,2,2],[1,1,2,3,3,2,4,20, 6,6,8,8,12,40,24,16,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,2,2,8,4,2,12,4,2,24,8, 4,4,4,4,12,8],[1,1,2,7,3,2,4,40,6,2,8,16,12,20,24,8,0,1,1,1,1,2,1,2,3,1, 2,4,1,2,6,1,2,8,2,3,2,2,4,4,2,4,12,1,4,24,2,6,4,1,6,8,2,8,12,2,12,4,2], [1,1,2,3,3,2,4,52,6,6,8,8,12,8,24,16,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2, 8,4,2,12,4,2,24,8,12,4,4],[1,1,2,3,3,2,4,36,6,6,8,24,12,24,0,1,1,1,1,2, 3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,8,4,12,4,2],[1,1,2,11,3,2,4,12,6,22, 8,8,12,24,24,16,0,1,1,1,1,2,1,1,3,2,1,4,2,1,6,2,1,12,4,2,2,1,2,4,1,2,6, 2,2,8,4,2,12,2,2,24,8,4,2,2,4,4,2,4,6,4,4,12,4],[1,1,2,15,3,2,4,32,6,18, 8,16,12,4,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8,2,3,4,2,4,2,2,4,12,1, 4,24,2,6,2,1,6,8,2,8,6,2,12,4,2],[1,1,2,27,3,2,4,28,6,6,8,8,12,8,24,16, 0,1,1,1,1,2,1,1,4,2,2,2,1,2,3,1,2,4,1,2,6,3,2,8,4,2,12,4,2,24,8,12,2,2, 12,4,2],[1,1,2,31,3,2,4,16,6,2,8,16,12,20,24,8,0,1,1,1,1,2,1,2,3,1,2,4, 1,2,6,1,2,8,2,3,4,2,4,4,2,4,12,1,4,24,2,6,2,1,6,8,2,8,12,2,12,2,2],[1,1, 2,23,3,2,4,24,6,10,8,16,12,12,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8, 2,3,4,2,4,2,1,4,4,1,4,12,1,4,24,2,6,2,1,6,8,2,8,6,1,8,12,1,12,2,1,12,4, 1],[1,1,2,19,3,2,4,20,6,14,8,24,12,16,0,1,1,1,1,2,1,1,4,2,2,2,1,2,3,1,2, 4,1,2,6,1,2,12,2,4,2,1,4,4,1,4,6,3,4,12,3,6,8,4,12,2,1,12,4,1],[1,1,2,11, 3,2,4,12,6,22,8,8,12,24,24,16,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,2,2,8,4,2,12, 4,2,24,8,4,2,2,4,4,2,4,6,4,4,12,4],[1,1,2,27,3,2,4,28,6,6,8,8,12,8,24,16, 0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,8,4,2,12,4,2,24,8,12,2,2,12,4,2],[1,1, 2,11,3,2,4,28,6,22,8,24,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,2,2,4,6, 4,4,12,2,6,8,4,12,4,2],[1,1,2,27,3,2,4,12,6,6,8,24,12,24,0,1,1,1,1,2,3, 2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,8,4,12,2,2],[1,1,2,3,3,2,4,20,6,6,8,8, 12,40,24,16,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,2,2,8,4,2,12,4,2,24,8,4,4,4,4, 12,8],[1,1,2,3,3,2,4,52,6,6,8,8,12,8,24,16,0,1,1,1,1,2,3,2,3,1,2,4,2,2, 6,3,2,8,4,2,12,4,2,24,8,12,4,4],[1,1,2,3,3,2,4,36,6,6,8,24,12,24,0,1,1, 1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,8,4,12,4,2],[1,1,2,3,3,2,4,36, 6,6,8,24,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,8,4,12,4, 2],[1,1,2,3,3,2,4,20,6,6,8,8,12,40,24,16,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4, 2,2,8,4,2,12,4,2,24,8,4,4,4,4,12,8],[1,1,2,3,3,2,4,52,6,6,8,8,12,8,24,16, 0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,8,4,2,12,4,2,24,8,12,4,4],[1,1,2,3,3, 2,4,36,6,6,8,24,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,8, 4,12,4,2],[1,1,2,3,3,2,4,20,6,6,8,8,12,40,24,16,0,1,1,1,1,2,3,1,3,2,1,6, 6,2,4,2,2,8,4,2,12,4,2,24,8,4,4,4,4,12,8],[1,1,2,3,3,2,4,52,6,6,8,8,12, 8,24,16,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,2,8,4,2,12,4,2,24,8,12,4,4],[1, 1,2,3,3,2,4,36,6,6,8,24,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12, 6,6,8,4,12,4,2],[1,1,2,7,3,2,4,16,6,14,8,8,12,32,24,16,0,1,1,1,1,2,1,1, 3,2,1,4,2,1,6,2,1,12,4,2,2,1,2,4,5,2,6,2,2,12,10,4,2,1,4,4,1,4,6,2,4,8, 2,4,12,2,4,24,4],[1,1,2,15,3,2,4,40,6,6,8,8,12,8,24,16,0,1,1,1,1,2,1,1, 4,2,2,2,1,2,3,1,2,4,1,2,6,1,2,12,2,4,6,1,4,8,2,4,12,1,4,24,4,6,4,4,12,2, 1,12,4,1],[1,1,2,7,3,2,4,32,6,14,8,24,12,16,0,1,1,1,1,2,1,1,4,2,2,2,1,2, 3,1,2,4,1,2,6,1,2,12,2,4,2,1,4,4,1,4,6,3,4,12,3,6,4,4,12,8,2],[1,1,2,15, 3,2,4,24,6,6,8,24,12,24,0,1,1,1,1,2,1,1,4,2,2,2,1,2,3,1,2,4,5,2,6,3,2,12, 12,12,2,1,12,4,1,12,8,2],[1,1,2,3,3,2,4,4,6,6,8,24,12,8,24,48,0,1,1,1,1, 2,1,1,3,2,1,4,2,1,6,2,1,12,4,2,2,1,2,4,1,2,6,2,2,8,4,2,12,2,2,24,8,4,8, 4,4,24,8],[1,1,2,3,3,2,4,4,6,6,8,56,12,8,24,16,0,1,1,1,1,2,1,1,4,2,2,2, 1,2,3,1,2,4,1,2,6,3,2,8,4,2,12,4,2,24,8,12,8,4],[1,1,2,3,3,2,4,4,6,6,8, 56,12,8,24,16,0,1,1,1,1,2,1,1,4,2,2,2,1,2,3,1,2,4,1,2,6,1,2,12,2,4,6,1, 4,8,2,4,12,1,4,24,4,6,8,4,12,8,2],[1,1,2,19,3,2,4,12,6,38,12,24,0,1,1,1, 1,2,3,1,3,2,1,6,6,2,2,6,2,6,12,4,2,1,4,4,3,4,6,2,4,12,6],[1,1,2,35,3,2, 4,28,6,22,12,8,0,1,1,1,1,2,3,2,2,2,2,3,1,2,6,3,4,2,1,4,4,1,4,6,4,4,12,2, 6,2,4,12,4,2],[1,1,2,43,3,2,4,20,6,14,12,16,0,1,1,1,1,2,3,2,2,2,2,3,1,2, 6,3,4,4,2,4,6,2,4,12,4,6,2,4,12,2,1,12,4,1],[1,1,2,27,3,2,4,36,6,30,0,1, 1,1,1,2,3,2,2,6,2,3,1,2,6,15,12,2,1,12,4,3],[1,1,2,19,3,32,4,12,6,32,0, 1,1,1,3,2,1,4,2,1,6,2,2,12,4,1,16,3,2,16,6,2],[1,1,2,19,3,2,4,12,6,38,12, 24,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,6,2,12,12,4,2,4,4,6,8],[1,1,2,35,3,2,4, 28,6,22,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,2,2,4,6,4,4,12,2,6,4,4,12, 2,2],[1,1,2,51,3,2,4,12,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,2,12, 12,12,2,4],[1,1,2,19,3,32,4,12,6,32,0,1,1,1,3,2,1,4,2,1,6,4,2,12,2,1,16, 3,2,16,6,2],[1,1,2,3,3,2,4,28,6,6,12,56,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,6, 2,12,12,4,4,4,4,12,8],[1,1,2,3,3,2,4,60,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2, 4,2,2,6,3,4,4,2,4,12,6,6,4,4,12,4,2],[1,1,2,3,3,2,4,60,6,6,12,24,0,1,1, 1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,4,4,12,4,2],[1,1,2,15,3,2,4,16, 6,30,12,32,0,1,1,1,1,2,3,1,3,2,1,6,6,2,2,2,2,4,4,2,6,4,2,12,8,4,2,2,4,4, 2,4,6,4,4,12,4],[1,1,2,31,3,2,4,32,6,14,12,16,0,1,1,1,1,2,3,2,2,2,2,3,1, 2,4,4,2,6,7,2,12,8,12,2,2,12,4,2],[1,1,2,23,3,2,4,40,6,22,12,8,0,1,1,1, 1,2,3,2,3,1,2,4,2,2,6,3,4,2,2,4,6,4,4,12,2,6,2,2,6,4,2,12,4,2],[1,1,2,39, 3,2,4,24,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,2,2,6, 4,2,12,2,2],[1,1,2,31,3,2,4,32,6,14,12,16,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6, 3,4,2,1,4,4,1,4,6,2,4,12,4,6,2,2,6,4,2,12,2,1,12,4,1],[1,1,2,23,3,2,4,40, 6,22,12,8,0,1,1,1,1,2,3,2,2,2,2,3,1,2,6,3,4,2,1,4,4,1,4,6,4,4,12,2,6,4, 4,12,2,1,12,4,1],[1,1,2,7,3,2,4,24,6,14,12,48,0,1,1,1,1,2,3,1,3,2,1,6,6, 2,2,2,2,4,4,2,6,4,2,12,8,4,4,4,4,12,8],[1,1,2,7,3,2,4,56,6,14,12,16,0,1, 1,1,1,2,3,2,2,2,2,3,1,2,4,4,2,6,7,2,12,8,12,4,4],[1,1,2,15,3,2,4,48,6,6, 12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,2,2,6,4,2,12,4,2], [1,1,2,15,3,2,4,48,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12, 6,6,2,2,6,4,2,12,4,2],[1,1,2,7,3,2,4,56,6,14,12,16,0,1,1,1,1,2,3,2,2,2, 2,3,1,2,6,3,4,4,2,4,6,2,4,12,4,6,4,4,12,4,2],[1,1,2,15,3,2,4,48,6,6,12, 24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,2,2,6,4,2,12,4,2],[1, 1,2,11,3,2,4,20,6,22,12,40,0,1,1,1,1,2,3,1,3,2,1,6,6,2,2,2,2,4,4,2,6,4, 2,12,8,4,2,1,4,4,3,4,6,2,4,12,6],[1,1,2,19,3,2,4,44,6,14,12,16,0,1,1,1, 1,2,3,2,2,2,2,3,1,2,4,4,2,6,7,2,12,8,12,2,1,12,4,3],[1,1,2,19,3,2,4,44, 6,14,12,16,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,2,1,4,4,1,4,6,2,4,12,4,6,2, 2,6,4,2,12,4,2],[1,1,2,27,3,2,4,36,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2, 2,6,3,4,4,2,4,12,6,6,2,2,6,4,2,12,2,1,12,4,1],[1,1,2,11,3,2,4,52,6,22,12, 8,0,1,1,1,1,2,3,2,2,2,2,3,1,2,6,3,4,2,1,4,4,1,4,6,4,4,12,2,6,4,4,12,4,2], [1,1,2,19,3,2,4,44,6,14,12,16,0,1,1,1,1,2,3,2,2,2,2,3,1,2,6,3,4,4,2,4,6, 2,4,12,4,6,4,4,12,2,1,12,4,1],[1,1,2,11,3,2,4,20,6,22,12,40,0,1,1,1,1,2, 3,1,3,2,1,6,6,2,4,6,2,12,12,4,2,2,4,4,2,4,6,4,4,12,4],[1,1,2,11,3,2,4,52, 6,22,12,8,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4,2,2,4,6,4,4,12,2,6,4,4,12,4, 2],[1,1,2,27,3,2,4,36,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,2,12,12, 12,2,2,12,4,2],[1,1,2,27,3,2,4,36,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2, 6,3,4,4,2,4,12,6,6,4,4,12,2,2],[1,1,2,19,3,2,4,44,6,14,12,16,0,1,1,1,1, 2,3,2,3,1,2,4,2,2,6,3,4,2,1,4,4,1,4,6,2,4,12,4,6,4,4,12,2,1,12,4,1],[1, 1,2,3,3,2,4,28,6,6,12,56,0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,6,2,12,12,4,4,4, 4,12,8],[1,1,2,3,3,2,4,60,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,2,12, 12,12,4,4],[1,1,2,3,3,2,4,60,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3, 4,4,2,4,12,6,6,4,4,12,4,2],[1,1,2,3,3,2,4,60,6,6,12,24,0,1,1,1,1,2,3,2, 3,1,2,4,2,2,6,3,4,4,2,4,12,6,6,4,4,12,4,2],[1,1,2,7,3,2,4,24,6,14,12,48, 0,1,1,1,1,2,3,1,3,2,1,6,6,2,4,6,2,12,12,4,2,1,4,4,3,4,6,2,4,12,6],[1,1, 2,15,3,2,4,48,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,2,12,12,12,2,1, 12,4,3],[1,1,2,15,3,2,4,48,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,2,2,6,3,4, 4,2,4,12,6,6,4,4,12,2,1,12,4,1],[1,1,2,7,3,2,4,56,6,14,12,16,0,1,1,1,1, 2,3,2,3,1,2,4,2,2,6,3,4,2,1,4,4,1,4,6,2,4,12,4,6,4,4,12,4,2],[1,1,2,7,3, 32,4,24,6,32,0,1,1,1,3,2,1,4,2,1,6,4,2,12,4,1,16,3,2,16,6,2],[1,1,2,19, 3,2,4,12,6,38,12,24,0,1,1,1,1,2,1,1,3,2,1,6,2,2,2,9,2,4,6,2,6,18,2,12,12], [1,1,2,43,3,2,4,20,6,14,12,16,0,1,1,1,1,2,1,2,2,3,2,3,1,2,4,4,2,6,1,2,12, 2,4,6,3,4,12,3,6,2,6,6,4,2],[1,1,2,35,3,2,4,28,6,22,12,8,0,1,1,1,1,2,1, 2,2,5,2,3,1,2,4,2,2,6,3,4,6,4,4,12,2,6,2,4,6,4,4],[1,1,2,19,3,8,4,12,6, 8,12,48,0,1,1,1,1,2,1,2,4,3,4,3,2,4,6,2,6,2,3,6,4,1,8,12,6],[1,1,2,19,3, 32,4,12,6,32,0,1,1,1,1,2,1,6,2,3,6,4,2,16,3,2,16,6,2],[1,1,2,11,3,2,4,20, 6,22,12,40,0,1,1,1,1,2,1,1,3,2,1,6,2,2,2,5,2,4,10,2,6,10,2,12,20],[1,1, 2,19,3,2,4,44,6,14,12,16,0,1,1,1,1,2,1,2,2,3,2,3,1,2,4,4,2,6,1,2,12,2,4, 6,3,4,12,3,6,2,2,6,4,6],[1,1,2,27,3,2,4,36,6,6,12,24,0,1,1,1,1,2,1,2,2, 1,2,3,1,2,4,6,2,6,3,4,12,6,6,2,4,6,4,4],[1,1,2,11,3,8,4,20,6,40,12,16,0, 1,1,1,1,2,1,2,2,2,2,4,1,4,3,2,4,6,2,6,2,1,6,4,3,8,6,4,8,12,2],[1,1,2,15, 3,2,4,8,6,30,8,8,12,16,24,16,0,1,1,1,1,2,1,1,3,2,1,6,2,2,2,1,2,4,2,2,6, 2,2,12,4,4,2,3,4,4,1,4,6,6,4,8,2,4,12,2,4,24,4],[1,1,2,39,3,2,4,16,6,6, 8,8,12,8,24,16,0,1,1,1,1,2,1,2,2,1,2,3,1,2,4,2,2,6,1,2,12,2,4,6,1,4,8,2, 4,12,1,4,24,4,12,2,3,12,4,1],[1,1,2,27,3,2,4,20,6,18,8,16,12,4,24,8,0,1, 1,1,1,2,1,2,3,1,2,4,1,2,6,1,4,2,2,4,8,1,4,12,1,4,24,2,6,2,1,6,4,1,8,6,2, 12,2,1,12,4,1,12,8,1],[1,1,2,35,3,2,4,12,6,10,8,16,12,12,24,8,0,1,1,1,1, 2,1,2,3,1,2,4,1,2,6,1,4,2,1,4,4,1,4,8,1,4,12,1,4,24,2,6,2,1,6,4,1,8,6,1, 8,12,1,12,2,2,12,8,1],[1,1,2,23,3,2,4,16,6,22,8,24,12,8,0,1,1,1,1,2,1,2, 2,1,2,3,1,2,4,2,2,6,3,4,2,2,4,6,4,4,12,2,12,2,1,12,4,1,12,8,2],[1,1,2,31, 3,2,4,8,6,14,8,24,12,16,0,1,1,1,1,2,1,2,2,1,2,3,1,2,4,2,2,6,1,2,12,2,4, 2,1,4,4,1,4,6,3,4,12,3,12,2,2,12,8,2],[1,1,2,7,3,2,4,16,6,14,8,8,12,32, 24,16,0,1,1,1,1,2,1,1,3,2,1,6,2,2,2,1,2,4,2,2,6,2,2,12,4,4,2,1,4,4,3,4, 6,2,4,8,2,4,12,6,4,24,4],[1,1,2,15,3,2,4,40,6,6,8,8,12,8,24,16,0,1,1,1, 1,2,1,2,2,1,2,3,1,2,4,2,2,6,1,2,12,2,4,6,1,4,8,2,4,12,1,4,24,4,12,2,1,12, 4,3],[1,1,2,11,3,2,4,36,6,10,8,16,12,12,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1, 2,6,1,4,2,1,4,4,1,4,8,1,4,12,1,4,24,2,6,2,1,6,4,1,8,6,1,8,12,1,12,4,2,12, 8,1],[1,1,2,19,3,2,4,28,6,2,8,16,12,20,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2, 6,1,4,4,2,4,8,1,4,12,1,4,24,2,6,2,1,6,4,1,8,12,2,12,2,1,12,4,1,12,8,1], [1,1,2,7,3,2,4,32,6,14,8,24,12,16,0,1,1,1,1,2,1,2,2,1,2,3,1,2,4,2,2,6,1, 2,12,2,4,2,1,4,4,1,4,6,3,4,12,3,12,4,2,12,8,2],[1,1,2,15,3,2,4,24,6,6,8, 24,12,24,0,1,1,1,1,2,1,2,2,1,2,3,1,2,4,2,2,6,3,4,4,2,4,12,6,12,2,1,12,4, 1,12,8,2],[1,1,2,11,3,2,4,20,6,22,12,40,0,1,1,1,1,2,1,1,3,2,1,6,2,2,2,3, 2,6,6,4,2,1,4,4,5,4,6,2,4,12,10],[1,1,2,19,3,2,4,44,6,14,12,16,0,1,1,1, 1,2,1,2,2,3,2,3,1,2,6,3,4,4,2,4,6,2,4,12,4,12,2,1,12,4,3],[1,1,2,11,3,2, 4,52,6,22,12,8,0,1,1,1,1,2,1,2,2,3,2,3,1,2,6,3,4,2,1,4,4,1,4,6,4,4,12,2, 12,4,4],[1,1,2,11,3,2,4,4,6,22,8,16,12,8,24,32,0,1,1,1,1,2,1,1,3,2,1,6, 2,2,2,1,2,4,2,2,6,2,2,12,4,4,2,2,4,6,4,4,8,4,4,24,8],[1,1,2,27,3,2,4,4, 6,6,8,32,12,8,24,16,0,1,1,1,1,2,1,2,2,1,2,3,1,2,4,2,2,6,1,2,12,2,4,6,1, 4,8,2,4,12,1,4,24,4,12,2,2,12,8,2],[1,1,2,11,3,2,4,4,6,22,8,48,12,8,0,1, 1,1,1,2,1,2,2,1,2,3,1,2,4,2,2,6,3,4,2,2,4,6,4,4,12,2,12,8,4],[1,1,2,3,3, 2,4,12,6,6,8,16,12,24,24,32,0,1,1,1,1,2,1,1,3,2,1,6,2,2,2,1,2,4,2,2,6,2, 2,12,4,4,4,2,4,8,4,4,12,4,4,24,8],[1,1,2,3,3,2,4,28,6,6,8,32,12,8,24,16, 0,1,1,1,1,2,1,2,2,1,2,3,1,2,4,2,2,6,1,2,12,2,4,6,1,4,8,2,4,12,1,4,24,4, 12,4,2,12,8,2],[1,1,2,3,3,2,4,12,6,6,8,48,12,24,0,1,1,1,1,2,1,2,2,1,2,3, 1,2,4,2,2,6,3,4,4,2,4,12,6,12,8,4],[1,1,2,17,3,2,4,2,6,34,8,4,12,4,16,8, 24,8,48,16,0,1,1,1,1,2,1,1,3,2,1,6,2,2,4,1,2,8,2,2,12,2,2,16,4,2,24,4,2, 48,8,8,2,2,8,6,4],[1,1,2,49,3,2,4,2,6,2,8,4,12,4,16,8,24,8,48,16,0,1,1, 1,1,2,1,2,3,1,2,4,1,2,6,1,2,8,2,2,12,2,2,16,4,2,24,4,2,48,8,24,2,2],[1, 1,2,33,3,2,4,2,6,18,8,4,12,4,16,24,24,8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1, 2,8,2,4,12,1,4,24,2,6,16,4,8,2,1,8,6,2,24,2,1],[1,1,2,9,3,2,4,10,6,18,8, 4,12,20,16,8,24,8,48,16,0,1,1,1,1,2,1,1,3,2,1,6,2,2,4,1,2,8,2,2,12,2,2, 16,4,2,24,4,2,48,8,8,2,1,8,4,1,8,6,2,8,12,2],[1,1,2,25,3,2,4,26,6,2,8,4, 12,4,16,8,24,8,48,16,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8,2,2,12,2,2,16, 4,2,24,4,2,48,8,24,2,1,24,4,1],[1,1,2,9,3,2,4,26,6,18,8,4,12,4,16,24,24, 8,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8,2,4,12,1,4,24,2,6,16,4,8,2,1,8,6, 2,24,4,1],[1,1,2,25,3,2,4,10,6,2,8,4,12,20,16,24,24,8,0,1,1,1,1,2,1,2,3, 1,2,4,1,2,6,1,2,8,2,4,12,1,4,24,2,6,16,4,8,4,1,8,12,2,24,2,1],[1,1,2,1, 3,2,4,18,6,2,8,4,12,36,16,8,24,8,48,16,0,1,1,1,1,2,1,1,3,2,1,6,2,2,4,1, 2,8,2,2,12,2,2,16,4,2,24,4,2,48,8,8,4,2,8,12,4],[1,1,2,1,3,2,4,50,6,2,8, 4,12,4,16,8,24,8,48,16,0,1,1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8,2,2,12,2,2,16, 4,2,24,4,2,48,8,24,4,2],[1,1,2,1,3,2,4,34,6,2,8,4,12,20,16,24,24,8,0,1, 1,1,1,2,1,2,3,1,2,4,1,2,6,1,2,8,2,4,12,1,4,24,2,6,16,4,8,4,1,8,12,2,24, 4,1],[1,1,2,27,3,8,4,12,6,24,8,24,0,1,1,1,1,2,3,6,4,2,6,8,4,8,3,1,8,6,3, 12,2,2],[1,1,2,3,3,8,4,36,6,24,8,24,0,1,1,1,1,2,3,6,4,2,6,8,4,8,3,1,8,6, 3,12,4,2],[1,1,2,3,3,8,4,36,6,24,8,24,0,1,1,1,1,2,3,6,4,2,6,8,4,8,3,1,8, 6,3,12,4,2],[1,1,2,15,3,8,4,24,6,24,8,24,0,1,1,1,1,2,1,2,2,1,6,4,2,8,3, 1,8,6,3,12,2,1,12,4,1,12,8,2],[1,1,2,19,3,8,4,20,6,8,8,24,12,16,0,1,1,1, 1,2,1,1,4,2,6,2,1,6,4,1,6,8,4,8,3,1,8,6,1,8,12,2,12,2,1,12,4,1],[1,1,2, 7,3,8,4,32,6,8,8,24,12,16,0,1,1,1,1,2,1,1,4,2,6,2,1,6,4,5,8,3,1,8,6,1,8, 12,2,12,8,2],[1,1,2,31,3,8,4,8,6,8,8,24,12,16,0,1,1,1,1,2,1,2,4,1,6,2,1, 6,4,1,8,3,1,8,6,1,8,12,2,12,2,2,12,8,2],[1,1,2,7,3,8,4,32,6,8,8,24,12,16, 0,1,1,1,1,2,1,2,4,1,6,2,1,6,4,1,8,3,1,8,6,1,8,12,2,12,4,2,12,8,2],[1,1, 2,15,3,32,4,24,8,24,0,1,1,1,3,2,1,3,4,2,6,4,1,12,2,1,12,4,1,12,8,2,32,3, 1],[1,1,2,19,3,8,4,44,6,8,12,16,0,1,1,1,1,2,1,1,4,2,3,2,2,3,4,2,6,2,2,6, 4,6,8,3,1,8,6,1,8,12,2],[1,1,2,7,3,8,4,8,6,8,8,48,12,16,0,1,1,1,1,2,1,1, 4,2,3,2,2,3,4,2,6,8,8,8,3,1,8,6,1,8,12,2],[1,1,2,31,3,8,4,32,6,8,12,16, 0,1,1,1,1,2,1,2,4,1,3,2,2,6,4,1,8,3,1,8,6,1,8,12,2,12,2,2,12,4,2],[1,1, 2,7,3,8,4,56,6,8,12,16,0,1,1,1,1,2,1,2,4,1,3,2,2,6,4,1,8,3,1,8,6,1,8,12, 2,12,4,4],[1,1,2,39,3,8,4,24,6,24,0,1,1,1,1,2,3,3,2,4,6,2,4,6,4,4,8,3,1, 8,6,3],[1,1,2,15,3,8,4,48,6,24,0,1,1,1,1,2,3,3,2,4,6,4,8,8,3,1,8,6,3],[1, 1,2,27,3,8,4,36,6,24,0,1,1,1,1,2,1,2,2,1,3,2,2,6,2,1,8,3,1,8,6,3,12,2,1, 12,4,3],[1,1,2,27,3,32,4,36,0,1,1,1,3,2,3,6,2,1,12,2,1,12,4,3,32,3,1],[1, 1,2,3,3,2,4,60,6,6,12,24,0,1,1,1,1,2,3,2,3,1,2,4,6,2,6,3,2,12,12,12,4,4]]; GroupIdOps.GroupSquares96:=[,,,,,,,,,,,,,,,,,,,,,,,,,,,,, [4,[30,46,75],[[24],[8,16],[8,8,8]]], [4,[31,47,49,76],[[56],[8,48],[24,32],[8,24,24]]], [4,[32,48],[[48],[16,32]]],,, [4,[35,77],[[24],[8,8,8]]],,,,,,,,,,,,,,, [8,[50,81],[[8,8],[4,4,4,4]]], [8,[51,82],[[16,16],[4,4,12,12]]],, [8,[53,83],[[24,24],[12,12,12,12]]],,,,,,,,, [4,[62,63],[[12,12,36],[12,20,28]]],,,,,,,,,,,,,,,, [8,[78,84],[[4,4,4,4],[8,8]]], [8,[79,86],[[4,4,12,12],[16,16]]], [8,[80,85],[[12,12,12,12],[24,24]]],,,,,,,,,,,,,, [4,[94,109],[[12],[4,8]]], [4,[95,110],[[28],[4,24]]],, [4,[97,111],[[28],[4,24]]], [4,[98,112],[[12],[4,8]]], [4,[99,113,117,120],[[20],[8,12],[4,16],[4,8,8]]],, [4,[101,114,118,121],[[52],[24,28],[4,48],[4,24,24]]], [4,[102,115,116,119,122],[[36],[12,24],[8,28], [4,32],[4,8,24]]],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, [4,[139,165],[[4,4,20],[4,12,12]]], [4,[140,141,167,168], [[12,12,36],[4,12,44],[4,20,36],[12,20,28]]],,,,,,,,,, [4,[150,151,153],[[24,24],[8,40],[16,32]]],,,,,,,,,,,,,,,, [4,[166,231],[[4,28,28],[4,4,52]]], ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, [4,[215,216],[[36],[12,24]]]]; ############################################################################# ## #F GroupIdOps.Group98( ) . . . . . . . . . . . . . . groups of order 98 ## GroupIdOps.Group98 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList98, x ); # and return return rec( catalogue := [ 98, nr ], names := GroupIdOps.GroupName98[nr], size := 98 ); end; GroupIdOps.GroupName98 := [ [ "14x7" ], [ "98" ], [ "D98" ], [ "7xD14" ], [ "7^2:2" ] ]; GroupIdOps.GroupList98 := [ [ 1, 1, 2, 1, 7, 48, 14, 48 ], [ 1, 1, 2, 1, 7, 6, 14, 6, 49, 42, 98, 42 ], [ 1, 1, 2, 49, 7, 6, 49, 42 ], [ 1, 1, 2, 7, 7, 48, 14, 42 ], [ 1, 1, 2, 49, 7, 48 ] ]; ############################################################################# ## #F GroupIdOps.Group100( ) . . . . . . . . . . . . . groups of order 100 ## GroupIdOps.Group100 := function( G ) local x, nr; # look up element orders in group list x := Flat( Collected( List( Elements(G), y -> Order(G,y) ) ) ); nr := Position( GroupIdOps.GroupList100, x ); # problem pair 13/14, 100.13 = 10, 100.14 = 6 if nr = 13 and Length(NormalSubgroups(G)) = 6 then nr := 14; fi; # and return return rec( catalogue := [ 100, nr ], names := GroupIdOps.GroupName100[nr], size := 100 ); end; GroupIdOps.GroupName100 := [ ["10^2"], ["50x2"], ["5x20"], ["100"], ["10xD10"], ["5x10.2"], ["5x5:4"], ["2x5^2:2"], ["(5^2x2).2"], ["D100"], ["50.2"], ["D10^2"], [], [], [] ]; GroupIdOps.GroupList100 := [ [ 1, 1, 2, 3, 5, 24, 10, 72 ], [ 1, 1, 2, 3, 5, 4, 10, 12, 25, 20, 50, 60 ], # 100.3 [ 1, 1, 2, 1, 4, 2, 5, 24, 10, 24, 20, 48 ], # 100.2 [ 1, 1, 2, 1, 4, 2, 5, 4, 10, 4, 20, 8, 25, 20, 50, 20, 100, 40 ], [ 1, 1, 2, 11, 5, 24, 10, 64 ], [ 1, 1, 2, 1, 4, 10, 5, 24, 10, 24, 20, 40 ], [ 1, 1, 2, 5, 4, 10, 5, 24, 10, 20, 20, 40 ], [ 1, 1, 2, 51, 5, 24, 10, 24 ], [ 1, 1, 2, 1, 4, 50, 5, 24, 10, 24 ], [ 1, 1, 2, 51, 5, 4, 10, 4, 25, 20, 50, 20 ], [ 1, 1, 2, 1, 4, 50, 5, 4, 10, 4, 25, 20, 50, 20 ], [ 1, 1, 2, 35, 5, 24, 10, 40 ], [ 1, 1, 2, 25, 4, 50, 5, 24 ], # 100.13 looks like 100.14 [ 1, 1, 2, 25, 4, 50, 5, 24 ], [ 1, 1, 2, 5, 4, 50, 5, 24, 10, 20 ], [ 1, 1, 2, 25, 4, 50, 5, 4, 25, 20 ] ]; ############################################################################# ## #F GroupIdOps.GroupPQ( ) . . . . . . . . non-abelian groups of order pq ## GroupIdOps.GroupPQ := function( G ) local n, name, p; # if size is 6, it S3 = D6 n := Size(G); p := Factors(n); if n = 6 then name := [ "D6", "S3" ]; # p = 2 call our group "D_pq" elif n mod 2 = 0 then name := [ Concatenation( "D", StringInt(n) ) ]; if IsPrimePowerInt(n/2+1) then Add( name, Concatenation( "O+_2(", StringInt(n/2+1), ")" ) ); fi; if IsPrimePowerInt(n/2-1) then Add( name, Concatenation( "O-_2(", StringInt(n/2-1), ")" ) ); fi; # otherwise call it "q:p" else name := [ Concatenation( StringInt(p[2]), ":", StringInt(p[1]) ) ]; fi; # this is always the second group in the catalogue (cyclic is first) List( name, IsString ); return rec( catalogue := [ n, 2 ], size := n, 3primes := [ "D", p[1], p[2] ], names := name ); end; ############################################################################# ## #F GroupIdOps.GroupP3( ) . . . . . . . . non-abelian groups of order p^3 ## GroupIdOps.GroupP3 := function( G ) local p, nr, type, names; # get prime p := Factors(Size(G))[1]; # at least p^2+1 elements of order p if NrElementsOfOrder( G, p ) > p^2 then nr := 4; type := [ "A", p ]; else nr := 5; type := [ "B", p ]; fi; # set name for small cases if p = 2 and nr = 4 then names := [ "D8" ]; elif p = 2 and nr = 5 then names := [ "Q8" ]; elif p = 3 and nr = 4 then names := [ "3^2:3" ]; elif p = 3 and nr = 5 then names := [ "9:3" ]; else names := []; fi; # and return return rec( catalogue := [ p^3, nr ], names := names, 3primes := type, size := p^3 ); end; ############################################################################# ## #F GroupIdOps.GroupP2Q( ) . . . . non-abelian groups of order p^2q, p 12 then Add( names, Concatenation( "H(", pn, "^2,", qn, ")" ) ); type := [ "H", p, q ]; fi; List( names, IsString ); # and return if IsBound(type) then return rec( catalogue := [ n, nr ], names := names, 3primes := type, size := n ); else return rec( catalogue := [ n, nr ], names := names, size := n ); fi; end; GroupIdOps.GroupNamesP2Q := []; GroupIdOps.GroupNamesP2Q[12] := [,,["D12"],["6.2"],["A4"] ]; GroupIdOps.GroupNamesP2Q[20] := [,,["D20"],["10.2"],["5:4"] ]; GroupIdOps.GroupNamesP2Q[28] := [,,["D28"],["14.2"] ]; GroupIdOps.GroupNamesP2Q[44] := [,,["D44"],["22.2"] ]; GroupIdOps.GroupNamesP2Q[52] := [,,["D52"],["26.2"],["13:4"] ]; GroupIdOps.GroupNamesP2Q[63] := [,,["7:3x3"],["21.3"] ]; GroupIdOps.GroupNamesP2Q[68] := [,,["D68"],["34.2"],["17:4"] ]; GroupIdOps.GroupNamesP2Q[76] := [,,["D76"],["38.2"] ]; GroupIdOps.GroupNamesP2Q[92] := [,,["D92"],["46.2"] ]; ############################################################################# ## #F GroupIdOps.GroupPQ2( ) . . . . non-abelian groups of order pq^2, p Size(Representative(t)) = q ); # if 1 = Length(Set(List(lat,t->Size(t)))) then syl := SylowSubgroup( G, q ); lat := List( ConjugacyClassesSubgroups(syl), Representative ); lat := Filtered( lat, t -> Size(t) = q ); nor := []; non := []; x := 1; while x <= Length(lat) and 0 = Length(non) and Length(nor) < 3 do if IsNormal( G, lat[x] ) then Add( nor, lat[x] ); else Add( non, lat[x] ); fi; x := x + 1; od; if 0 = Length(non) and 2 < Length(nor) then cat := [ "K", p, q ]; name := Concatenation( "K(", pn, ",", qn, "^2)" ); else while x <= Length(lat) and Length(nor) < 2 do if IsNormal( G, lat[x] ) then Add( nor, lat[x] ); fi; x := x + 1; od; A := nor[1].generators[1]; B := nor[2].generators[1]; C := SylowSubgroup( G, p ).generators[1]; AC := A^C; x := 1; PO := A^0; while PO <> AC do x := x+1; if x^p mod q = 1 then PO := A^x; fi; od; BC := B^C; s := 1; PO := B^0; while s < q and PO <> BC do s := s+1; if s mod p <> 0 and s mod p <> 1 then PO := B^((x^s) mod q); fi; od; s := s mod p; if ((1/s) mod p) < s then s := (1/s) mod p; fi; cat := [ "L", p, q, s ]; x := String(s); name := Concatenation( "L(", pn, ",", qn, "^2,", x, ")" ); fi; fi; # p | q+1, p <> 2 elif p <> 2 and (q+1) mod p = 0 then cat := [ "N", p, q ]; name := Concatenation( "N(", pn, ",", qn, "^2)" ); # this should not happen else Error( "you have encountered a bug" ); fi; IsString(name); # and return return rec( 3primes := cat, names := [name], size := n ); end; ############################################################################# ## #F GroupIdOps.GroupPQR( ) . . . non-abelian groups of order pqr, p IsNormal(G,t) ); # two normal sylow subgroup: Dpqxr or Gpqr(s) if nor = 2 then s1 := First( syl, t -> not IsNormal(G,t) ); syl := Filtered( syl, t -> t <> s1 ); s2 := Closure( s1, syl[1] ); s3 := Closure( s1, syl[2] ); # one product non-normal: Dpqxr syl := Filtered( [s2,s3], t -> IsNormal(G,t) ); if 1 = Length(syl) then p := Size(s1); q := Size(syl[1])/p; r := n/(p*q); if (q-1) mod p <> 0 then Error( "you have encountered a bug" ); fi; cat := [ "D", p, q, r ]; name := Concatenation( "D(", String(p), ",", String(q), ")x", String(r) ); # both non-normal: Gpqr(s) else p := Size(s1); q := Size(s2)/p; r := Size(s3)/p; if r < q then x := r; r := q; q := x; fi; if q = r or q = p or p = r then Error( "you have encountered a bug" ); fi; if (q-1) mod p <> 0 then Error( "you have encountered a bug" ); fi; if (r-1) mod p <> 0 then Error( "you have encountered a bug" ); fi; # find and C := SylowSubgroup(G,p).generators[1]; A := SylowSubgroup(G,r).generators[1]; AC := A^C; PO := A^0; s := 1; while AC <> PO do s := s+1; if s mod r <> 1 and s^p mod r = 1 then PO := A^s; fi; od; # correct x := First( [2..r-1], t -> t mod r <> 1 and t^p mod r = 1 ); s := LogMod( x, s, r ); C := C^s; # now find B := SylowSubgroup(G,q).generators[1]; BC := B^C; PO := B^0; s := 1; while BC <> PO do s := s+1; if s mod r <> 1 and s^p mod r = 1 then PO := B^s; fi; od; # and find y := First( [2..q-1], t -> t mod q <> 1 and t^p mod q = 1 ); s := LogMod( s, y, q ); # construct a name cat := [ "G", p, q, r, s ]; name := Concatenation( "G(", String(p), ",", String(q), ",", String(r), ",", String(s), ")" ); fi; # one normal sylow subgroup: Hpqr elif nor = 1 then cat := [ "H", p, q, r ]; name := Concatenation( "H(", String(p), ",", String(q), ",", String(r), ")" ); # this should not happen else Error( "you have encountered a bug" ); fi; # get name for small groups names := []; Add( names, name ); List( names, IsString ); # and return return rec( 3primes := cat, names := names, size := n ); end; ############################################################################# ## #F GroupIdOps.AbelianGroup1( ) . . . . . . . abelian groups of order p ## GroupIdOps.AbelianGroup1 := function( G ) local n; n := Size(G); return rec( catalogue := [ n, 1 ], size := n, names := [ StringInt(n) ] ); end; ############################################################################# ## #F GroupIdOps.AbelianGroup2( ) . . . . . . . abelian groups of order pq ## GroupIdOps.AbelianGroup2 := function( G ) local n, p, s, cat, inv, name; # get the order n := Size(G); p := Factors(n); s := List( p, StringInt ); # if p1 <> p2 then must be cyclic if p[1] <> p[2] then cat := [ n, 1 ]; inv := [ n ]; name := [ StringInt(n), Concatenation(s[1],"x",s[2]) ]; # p^2 has catalogue number 2 elif IsCyclic(G) then cat := [ n, 2 ]; inv := [ n ]; name := [ StringInt(n) ]; # p x p has catalogue number 1 else cat := [ n, 1 ]; inv := [ p[1], p[2] ]; name := [ Concatenation(s[1],"x",s[2]) ]; fi; # and return List( name, IsString ); return rec( catalogue := cat, size := n, names := name ); end; ############################################################################# ## #F GroupIdOps.AbelianGroup3( ) . . . . . . . abelian groups of order pqr ## GroupIdOps.AbelianGroup3 := function( G ) local i, n, p, s, d, inv, cat, name; # get the order n := Size(G); p := Factors(n); s := List( p, StringInt ); d := Set(p); # compute the abelian invariants inv := AbelianInvariants(G); # p x p x p has catalogue number 1 if Length(d) = 1 and Length(inv) = 3 then cat := [ n, 1 ]; name := [ Concatenation(s[1],"x",s[2],"x",s[3]) ]; # p x p^2 has catalogue number 2 elif Length(d) = 1 and Length(inv) = 2 then cat := [ n, 2 ]; name := [ Concatenation(s[1],"x",StringInt(p[2]*p[3])) ]; # p^3 has catalogue number 3 elif Length(d) = 1 and Length(inv) = 1 then cat := [ n, 3 ]; name := [ StringInt(n) ]; # p x p x q has catalogue number 1 elif Length(d) = 2 and Length(inv) = 3 then cat := [ n, 1 ]; if p[1] = p[2] then i := 3; else i := 1; fi; name := [ Concatenation(s[2],"x",s[2],"x",s[i]), Concatenation(s[2],"x",StringInt(p[2]*p[i])) ]; # p^2 x q has catalogue number 2 elif Length(d) = 2 and Length(inv) = 2 then cat := [ n, 2 ]; if p[1] = p[2] then i := 3; else i := 1; fi; name := [ Concatenation(StringInt(p[2]^2),"x",s[i]), StringInt(p[2]^2*p[i]) ]; # p x q x r has catalogue number 1 else cat := [ n, 1 ]; name := [ StringInt(n), Concatenation(s[1],"x",s[2],"x",s[3]) ]; fi; # and return List( name, IsString ); return rec( catalogue := cat, size := n, names := name ); end; ############################################################################# ## #F GroupIdOps.AbelianPGroup( ) . . . . . . . . . abelian 2- or 3-groups ## GroupIdOps.AbelianPGroup := function( G ) local n, inv, p; # check size n := Size(G); if not IsBound(GroupIdOps.AbelianPGroups[n]) then Error( " must be an abelian p-group of order at least 128" ); fi; # compute the abelian invariants inv := AbelianInvariants(G); Sort(inv); # and look them up p := First( GroupIdOps.AbelianPGroups[n], x -> x[1] = inv ); # and return return rec( size := n, pGroupId := p[2] ); end; ############################################################################# ## #F GroupIdOps.2Group( ) . . . . . . . . . 2-group of order at most 256 ## GroupIdOps.2Group := function ( G ) local size, sp, enc, total, name, i; # check the size size := Size(G); if not size in [2,4,8,16,32,64,128,256] then Error(" must be a power of 2 at most 256"); fi; # compute the standard presentation of sp := StandardPresentation( FpGroup(G), 2, "ClassBound", 8 ); # and encode it enc := EncodedStandardPresentation( sp, 2 ); # find the file total := 0; for name in RecFields(TGParts) do if size in TGParts.(name).sizes then TGLoad( TGParts.(name) ); for i in [ 1 .. TGParts.(name).number ] do if TGParts.(name).groups[i] = enc then return rec( size := size, pGroupId := i+total ); fi; od; total := total + TGParts.(name).number; fi; od; # if we end here something is wrong Error( "cannot find in 2-group library" ); end; ############################################################################# ## #F GroupIdOps.3Group( ) . . . . . . . . . 3-group of order at most 729 ## GroupIdOps.3Group := function ( G ) local size, sp, enc, total, name, i; # check the size size := Size(G); if not size in [3,9,27,81,243,729] then Error(" must be a power of 3 at most 729"); fi; # compute the standard presentation of sp := StandardPresentation( FpGroup(G), 3, "ClassBound", 6 ); # and encode it enc := EncodedStandardPresentation( sp, 3 ); # find the file total := 0; for name in RecFields(ThGParts) do if size in ThGParts.(name).sizes then ThGLoad( ThGParts.(name) ); for i in [ 1 .. ThGParts.(name).number ] do if ThGParts.(name).groups[i] = enc then return rec( size := size, pGroupId := i+total ); fi; od; total := total + ThGParts.(name).number; fi; od; # if we end here something is wrong Error( "cannot find in 3-group library" ); end; ############################################################################# ## #V GroupIdOps.NrSolvableGroups . . . . . number of sovable groups of order n ## GroupIdOps.NrSolvableGroups := [ 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, 1, 10, 1, 4, 2, 2, 1, 231, 1, 5, 2, 16 ]; ############################################################################# ## #F GroupId( ) . . . . . . . . . . . . . . . . find the catalogue id ## GroupId := function( G ) if not IsBound(G.groupId) then if not IsBound(G.operations.GroupId) then G.groupId := GroupOps.GroupId(G); else G.groupId := G.operations.GroupId(G); fi; fi; return G.groupId; end; GroupOps.GroupId := function( G ) local n, p, result, anupq; # get the size of and the prime factors n := Size(G); p := Factors(n); # check if n is too big and has more than three primes (doesn't work) anupq := [128,256,243,729]; if n > 100 and not n in anupq and Length(p) > 3 then Error( "size of must be less than 101 or a product of ", "at most 3 primes" ); fi; if n in anupq and not IsAbelian(G) and StandardPresentation = 0 then Error( "the ANU pq is required for 2- and 3-groups" ); fi; # handle abelian groups first if IsAbelian(G) then # for some order only one group of this order exsists (namely n) if n < 100 and GroupIdOps.NrSolvableGroups[n] = 1 then result := rec( catalogue := [ n, 1 ], names := [ StringInt(n) ], size := n ); # cases with at most 3 primes elif Length(p) = 1 then result := GroupIdOps.AbelianGroup1(G); elif Length(p) = 2 then result := GroupIdOps.AbelianGroup2(G); elif Length(p) = 3 then result := GroupIdOps.AbelianGroup3(G); # handle other cases elif n = 16 then result := GroupIdOps.Group16(G); elif n = 24 then result := GroupIdOps.Group24(G); elif n = 32 then result := GroupIdOps.Group32(G); elif n = 36 then result := GroupIdOps.Group36(G); elif n = 40 then result := GroupIdOps.Group40(G); elif n = 48 then result := GroupIdOps.Group48(G); elif n = 54 then result := GroupIdOps.Group54(G); elif n = 56 then result := GroupIdOps.Group56(G); elif n = 60 then result := GroupIdOps.Group60(G); elif n = 64 then result := GroupIdOps.Group64(G); elif n = 72 then result := GroupIdOps.Group72(G); elif n = 80 then result := GroupIdOps.Group80(G); elif n = 81 then result := GroupIdOps.Group81(G); elif n = 84 then result := GroupIdOps.Group84(G); elif n = 88 then result := GroupIdOps.Group88(G); elif n = 90 then result := GroupIdOps.Group90(G); elif n = 96 then result := GroupIdOps.Group96(G); elif n = 100 then result := GroupIdOps.Group100(G); elif n in anupq then result := GroupIdOps.AbelianPGroup(G); # return only the abelian invariants else result := rec( size := n ); fi; # add the abelian invariants and return result.abelianInvariants := AbelianInvariants(G); # if it is a p group check if know the number if IsPrimePowerInt(n) and IsBound(GroupIdOps.PGroups[n]) then result.pGroupId := GroupIdOps.PGroups[n][result.catalogue[2]]; fi; # and return return result; # non-abelian groups of order p*q elif Length(p) = 2 then result := GroupIdOps.GroupPQ(G); # non-abelian groups of types: p^3, p^2*q, q*p*r elif Length(p) = 3 then # type: p^3 if p[1] = p[2] and p[2] = p[3] then result := GroupIdOps.GroupP3(G); # type: p^2*q, p < q elif p[1] = p[2] and p[2] <> p[3] then result := GroupIdOps.GroupP2Q(G); # type: p*q^2, p < q elif 100 < n and p[1] <> p[2] and p[2] = p[3] then result := GroupIdOps.GroupPQ2(G); # type: p*q*r elif 100 < n and p[1]<>p[2] and p[2]<>p[3] then result := GroupIdOps.GroupPQR(G); # catch small cases missed above elif n = 18 then result := GroupIdOps.Group18(G); elif n = 50 then result := GroupIdOps.Group50(G); elif n = 75 then result := GroupIdOps.Group75(G); elif n = 98 then result := GroupIdOps.Group98(G); elif n = 30 then result := GroupIdOps.Group30(G); elif n = 42 then result := GroupIdOps.Group42(G); elif n = 66 then result := GroupIdOps.Group66(G); elif n = 70 then result := GroupIdOps.Group70(G); elif n = 78 then result := GroupIdOps.Group78(G); fi; # other small cases elif n = 16 then result := GroupIdOps.Group16(G); elif n = 24 then result := GroupIdOps.Group24(G); elif n = 32 then result := GroupIdOps.Group32(G); elif n = 36 then result := GroupIdOps.Group36(G); elif n = 40 then result := GroupIdOps.Group40(G); elif n = 48 then result := GroupIdOps.Group48(G); elif n = 54 then result := GroupIdOps.Group54(G); elif n = 56 then result := GroupIdOps.Group56(G); elif n = 60 then result := GroupIdOps.Group60(G); elif n = 64 then result := GroupIdOps.Group64(G); elif n = 72 then result := GroupIdOps.Group72(G); elif n = 80 then result := GroupIdOps.Group80(G); elif n = 81 then result := GroupIdOps.Group81(G); elif n = 84 then result := GroupIdOps.Group84(G); elif n = 88 then result := GroupIdOps.Group88(G); elif n = 90 then result := GroupIdOps.Group90(G); elif n = 96 then result := GroupIdOps.Group96(G); elif n = 100 then result := GroupIdOps.Group100(G); elif n in anupq and p[1] = 2 then result := GroupIdOps.2Group(G); elif n in anupq and p[1] = 3 then result := GroupIdOps.3Group(G); fi; # if it is a p group check if know the number if not IsBound(result.pGroupId) and IsBound(GroupIdOps.PGroups[n]) then result.pGroupId := GroupIdOps.PGroups[n][result.catalogue[2]]; result.pGroupId := ShallowCopy(result.pGroupId); fi; # if the order is a product of at most three primes check '3primes' if not IsBound(result.3primes) and IsBound(GroupIdOps.3primes[n]) then result.3primes := GroupIdOps.3primes[n][result.catalogue[2]]; result.3primes := ShallowCopy(result.3primes); fi; # and return return result; end;