# Group 64.54 downloaded from the LMFDB on 10 October 2026. ## Various presentations of this group are stored in this file: # GPC is polycyclic presentation, GPerm is permutation group # GLZ, GLFp, GLZN, GLZq, GLFq if they exist are matrix groups # Many characteristics of the group are stored as booleans in a record: # Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, # metacyclic, monomial, nilpotent, perfect, quasisimple, rational, # solvable, supersolvable # The character table is stored as a record chartbl_n_i where n is the order # of the group and i is which group of that order it is. The record is # converted to a character table using ConvertToLibraryCharacterTableNC # Constructions GPC := PcGroupCode(309504273051162771967,64); a := GPC.1; b := GPC.2; GPerm := Group( (1,34,2,33)(3,36,4,35)(5,38,6,37)(7,40,8,39)(9,42,10,41)(11,44,12,43)(13,46,14,45)(15,48,16,47)(17,50,18,49)(19,52,20,51)(21,54,22,53)(23,56,24,55)(25,58,26,57)(27,60,28,59)(29,62,30,61)(31,64,32,63), (1,18,2,17)(3,20,4,19)(5,22,6,21)(7,24,8,23)(9,26,10,25)(11,28,12,27)(13,30,14,29)(15,32,16,31)(33,58,34,57)(35,60,36,59)(37,62,38,61)(39,64,40,63)(41,56,42,55)(43,53,44,54)(45,50,46,49)(47,52,48,51), (1,13,5,12,4,16,8,10,2,14,6,11,3,15,7,9)(17,25,23,31,19,27,22,30,18,26,24,32,20,28,21,29)(33,41,39,47,35,43,38,46,34,42,40,48,36,44,37,45)(49,61,53,60,52,64,56,58,50,62,54,59,51,63,55,57), (1,7,3,6,2,8,4,5)(9,15,11,14,10,16,12,13)(17,21,20,24,18,22,19,23)(25,29,28,32,26,30,27,31)(33,37,36,40,34,38,35,39)(41,45,44,48,42,46,43,47)(49,55,51,54,50,56,52,53)(57,63,59,62,58,64,60,61), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,31,30,32)(33,35,34,36)(37,39,38,40)(41,43,42,44)(45,47,46,48)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64) ); GLFp := Group([[[ Z(31)^9, Z(31)^16 ], [ Z(31)^15, Z(31)^9 ]], [[ Z(31)^12, Z(31)^16 ], [ Z(31)^0, Z(31)^27 ]]]); # Booleans booleans_64_54 := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := true, metacyclic := true, monomial := true, nilpotent := true, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_64_54:=rec(); chartbl_64_54.IsFinite:= true; chartbl_64_54.UnderlyingCharacteristic:= 0; chartbl_64_54.UnderlyingGroup:= GPC; chartbl_64_54.Size:= 64; chartbl_64_54.InfoText:= "Character table for group 64.54 downloaded from the LMFDB."; chartbl_64_54.Identifier:= " Q64 "; chartbl_64_54.NrConjugacyClasses:= 19; chartbl_64_54.ConjugacyClasses:= [ of ..., f6, f5, f1*f2*f3*f6, f1*f5, f4, f4*f5, f3, f3*f4, f3*f5, f3*f4*f5, f2, f2*f3, f2*f4, f2*f5*f6, f2*f5, f2*f4*f6, f2*f4*f5, f2*f6]; chartbl_64_54.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]; chartbl_64_54.ComputedPowerMaps:= []; chartbl_64_54.ComputedPowerMaps[2]:= [1, 1, 2, 2, 2, 3, 3, 6, 7, 7, 6, 8, 9, 10, 11, 11, 10, 9, 8]; chartbl_64_54.SizesCentralizers:= [64, 64, 32, 4, 4, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32]; chartbl_64_54.ClassNames:= ["1A", "2A", "4A", "4B", "4C", "8A1", "8A3", "16A1", "16A3", "16A5", "16A7", "32A1", "32A3", "32A5", "32A7", "32A9", "32A11", "32A13", "32A15"]; chartbl_64_54.OrderClassRepresentatives:= [1, 2, 4, 4, 4, 8, 8, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32]; chartbl_64_54.Irr:= [[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1], [1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1], [2, 2, 2, 0, 0, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 0, 0, -2, -2, 0, 0, 0, 0, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, E(8)+E(8)^-1, -1*E(8)-E(8)^-1], [2, 2, 2, 0, 0, -2, -2, 0, 0, 0, 0, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, -1*E(8)-E(8)^-1, E(8)+E(8)^-1], [2, 2, -2, 0, 0, 0, 0, -1*E(16)^2-E(16)^-2, E(16)^2+E(16)^-2, E(16)^2+E(16)^-2, -1*E(16)^2-E(16)^-2, -1*E(16)^3-E(16)^-3, E(16)+E(16)^-1, -1*E(16)-E(16)^-1, E(16)^3+E(16)^-3, E(16)^3+E(16)^-3, -1*E(16)-E(16)^-1, E(16)+E(16)^-1, -1*E(16)^3-E(16)^-3], [2, 2, -2, 0, 0, 0, 0, -1*E(16)^2-E(16)^-2, E(16)^2+E(16)^-2, E(16)^2+E(16)^-2, -1*E(16)^2-E(16)^-2, E(16)^3+E(16)^-3, -1*E(16)-E(16)^-1, E(16)+E(16)^-1, -1*E(16)^3-E(16)^-3, -1*E(16)^3-E(16)^-3, E(16)+E(16)^-1, -1*E(16)-E(16)^-1, E(16)^3+E(16)^-3], [2, 2, -2, 0, 0, 0, 0, E(16)^2+E(16)^-2, -1*E(16)^2-E(16)^-2, -1*E(16)^2-E(16)^-2, E(16)^2+E(16)^-2, -1*E(16)-E(16)^-1, -1*E(16)^3-E(16)^-3, E(16)^3+E(16)^-3, E(16)+E(16)^-1, E(16)+E(16)^-1, E(16)^3+E(16)^-3, -1*E(16)^3-E(16)^-3, -1*E(16)-E(16)^-1], [2, 2, -2, 0, 0, 0, 0, E(16)^2+E(16)^-2, -1*E(16)^2-E(16)^-2, -1*E(16)^2-E(16)^-2, E(16)^2+E(16)^-2, E(16)+E(16)^-1, E(16)^3+E(16)^-3, -1*E(16)^3-E(16)^-3, -1*E(16)-E(16)^-1, -1*E(16)-E(16)^-1, -1*E(16)^3-E(16)^-3, E(16)^3+E(16)^-3, E(16)+E(16)^-1], [2, -2, 0, 0, 0, -1*E(32)^4-E(32)^-4, E(32)^4+E(32)^-4, -1*E(32)^6-E(32)^-6, E(32)^2+E(32)^-2, -1*E(32)^2-E(32)^-2, E(32)^6+E(32)^-6, -1*E(32)^5-E(32)^-5, E(32)+E(32)^-1, E(32)^7+E(32)^-7, E(32)^3+E(32)^-3, -1*E(32)^3-E(32)^-3, -1*E(32)^7-E(32)^-7, -1*E(32)-E(32)^-1, E(32)^5+E(32)^-5], [2, -2, 0, 0, 0, -1*E(32)^4-E(32)^-4, E(32)^4+E(32)^-4, -1*E(32)^6-E(32)^-6, E(32)^2+E(32)^-2, -1*E(32)^2-E(32)^-2, E(32)^6+E(32)^-6, E(32)^5+E(32)^-5, -1*E(32)-E(32)^-1, -1*E(32)^7-E(32)^-7, -1*E(32)^3-E(32)^-3, E(32)^3+E(32)^-3, E(32)^7+E(32)^-7, E(32)+E(32)^-1, -1*E(32)^5-E(32)^-5], [2, -2, 0, 0, 0, -1*E(32)^4-E(32)^-4, E(32)^4+E(32)^-4, E(32)^6+E(32)^-6, -1*E(32)^2-E(32)^-2, E(32)^2+E(32)^-2, -1*E(32)^6-E(32)^-6, -1*E(32)^3-E(32)^-3, E(32)^7+E(32)^-7, -1*E(32)-E(32)^-1, -1*E(32)^5-E(32)^-5, E(32)^5+E(32)^-5, E(32)+E(32)^-1, -1*E(32)^7-E(32)^-7, E(32)^3+E(32)^-3], [2, -2, 0, 0, 0, -1*E(32)^4-E(32)^-4, E(32)^4+E(32)^-4, E(32)^6+E(32)^-6, -1*E(32)^2-E(32)^-2, E(32)^2+E(32)^-2, -1*E(32)^6-E(32)^-6, E(32)^3+E(32)^-3, -1*E(32)^7-E(32)^-7, E(32)+E(32)^-1, E(32)^5+E(32)^-5, -1*E(32)^5-E(32)^-5, -1*E(32)-E(32)^-1, E(32)^7+E(32)^-7, -1*E(32)^3-E(32)^-3], [2, -2, 0, 0, 0, E(32)^4+E(32)^-4, -1*E(32)^4-E(32)^-4, -1*E(32)^2-E(32)^-2, -1*E(32)^6-E(32)^-6, E(32)^6+E(32)^-6, E(32)^2+E(32)^-2, -1*E(32)^7-E(32)^-7, E(32)^5+E(32)^-5, E(32)^3+E(32)^-3, -1*E(32)-E(32)^-1, E(32)+E(32)^-1, -1*E(32)^3-E(32)^-3, -1*E(32)^5-E(32)^-5, E(32)^7+E(32)^-7], [2, -2, 0, 0, 0, E(32)^4+E(32)^-4, -1*E(32)^4-E(32)^-4, -1*E(32)^2-E(32)^-2, -1*E(32)^6-E(32)^-6, E(32)^6+E(32)^-6, E(32)^2+E(32)^-2, E(32)^7+E(32)^-7, -1*E(32)^5-E(32)^-5, -1*E(32)^3-E(32)^-3, E(32)+E(32)^-1, -1*E(32)-E(32)^-1, E(32)^3+E(32)^-3, E(32)^5+E(32)^-5, -1*E(32)^7-E(32)^-7], [2, -2, 0, 0, 0, E(32)^4+E(32)^-4, -1*E(32)^4-E(32)^-4, E(32)^2+E(32)^-2, E(32)^6+E(32)^-6, -1*E(32)^6-E(32)^-6, -1*E(32)^2-E(32)^-2, -1*E(32)-E(32)^-1, -1*E(32)^3-E(32)^-3, E(32)^5+E(32)^-5, E(32)^7+E(32)^-7, -1*E(32)^7-E(32)^-7, -1*E(32)^5-E(32)^-5, E(32)^3+E(32)^-3, E(32)+E(32)^-1], [2, -2, 0, 0, 0, E(32)^4+E(32)^-4, -1*E(32)^4-E(32)^-4, E(32)^2+E(32)^-2, E(32)^6+E(32)^-6, -1*E(32)^6-E(32)^-6, -1*E(32)^2-E(32)^-2, E(32)+E(32)^-1, E(32)^3+E(32)^-3, -1*E(32)^5-E(32)^-5, -1*E(32)^7-E(32)^-7, E(32)^7+E(32)^-7, E(32)^5+E(32)^-5, -1*E(32)^3-E(32)^-3, -1*E(32)-E(32)^-1]]; ConvertToLibraryCharacterTableNC(chartbl_64_54);