# Group 1024.dhg downloaded from the LMFDB on 04 October 2025. ## Various presentations of this group are stored in this file: # GPC is polycyclic presentation GPerm is permutation group # GLZ, GLFp, GLZA, 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(30383902621016833058827572899707802992256306489651377678546313046874520519777039793,1024); a := GPC.1; b := GPC.3; c := GPC.4; d := GPC.5; e := GPC.7; f := GPC.8; GPerm := Group( (1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16), (1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15) ); # Booleans booleans_1024_dhg := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := false, metacyclic := false, monomial := true, nilpotent := true, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_1024_dhg:=rec(); chartbl_1024_dhg.IsFinite:= true; chartbl_1024_dhg.UnderlyingCharacteristic:= 0; chartbl_1024_dhg.UnderlyingGroup:= GPerm; chartbl_1024_dhg.Size:= 1024; chartbl_1024_dhg.InfoText:= "Character table for group 1024.dhg downloaded from the LMFDB."; chartbl_1024_dhg.Identifier:= " C2^6.SD16 "; chartbl_1024_dhg.NrConjugacyClasses:= 34; chartbl_1024_dhg.ConjugacyClasses:= [(), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (9,10)(11,12)(13,14)(15,16), (11,12)(15,16), (3,4)(7,8)(11,12)(15,16), (3,4)(7,8)(9,10)(11,12)(13,14)(15,16), (5,6)(7,8)(11,12)(15,16), (5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (13,14)(15,16), (3,4)(7,8)(13,14)(15,16), (5,6)(7,8)(13,14)(15,16), (1,6)(2,5)(3,8)(4,7)(9,14)(10,13)(11,16)(12,15), (1,6,2,5)(3,7,4,8)(9,14,10,13)(11,15,12,16), (1,6,2,5)(3,8,4,7)(9,14,10,13)(11,15,12,16), (1,5)(2,6)(3,7)(4,8)(9,13,10,14)(11,15,12,16), (3,7)(4,8)(9,11)(10,12)(13,15,14,16), (3,7)(4,8)(9,11)(10,12)(13,16,14,15), (3,7,4,8)(5,6)(9,11)(10,12)(13,15,14,16), (3,7,4,8)(5,6)(9,11)(10,12)(13,16,14,15), (3,7,4,8)(5,6)(9,11,10,12)(13,15)(14,16), (3,7,4,8)(5,6)(9,11,10,12)(13,16)(14,15), (1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,15,14,16), (1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,16,14,15), (1,10,6,13)(2,9,5,14)(3,16,8,11)(4,15,7,12), (1,13,6,10)(2,14,5,9)(3,11,8,16)(4,12,7,15), (1,7,6,4,2,8,5,3)(9,15,14,12,10,16,13,11), (1,3,5,8,2,4,6,7)(9,11,13,16,10,12,14,15), (1,3,6,8,2,4,5,7)(9,11,14,15,10,12,13,16), (1,13,6,10,2,14,5,9)(3,12,8,15,4,11,7,16), (1,12,6,16,2,11,5,15)(3,10,8,13,4,9,7,14), (1,15,7,14,6,12,4,10,2,16,8,13,5,11,3,9), (1,9,3,11,5,13,8,16,2,10,4,12,6,14,7,15), (1,14,4,16,5,9,7,12,2,13,3,15,6,10,8,11), (1,11,8,10,6,15,3,13,2,12,7,9,5,16,4,14)]; chartbl_1024_dhg.IdentificationOfConjugacyClasses:= [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]; chartbl_1024_dhg.ComputedPowerMaps:= [ , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 9, 9, 10, 10, 10, 10, 9, 9, 12, 12, 13, 13, 14, 14, 14, 26, 27, 27, 26]]; chartbl_1024_dhg.SizesCentralizers:= [1024, 1024, 512, 256, 256, 256, 128, 128, 128, 128, 64, 64, 128, 128, 32, 32, 32, 32, 32, 32, 32, 32, 32, 16, 16, 32, 32, 16, 16, 16, 16, 16, 16, 16]; chartbl_1024_dhg.ClassNames:= ["1A", "2A", "2B", "2C", "2D", "2E", "2F", "2G", "2H", "2I", "2J", "2K", "4A", "4B", "4C", "4D1", "4D-1", "4E1", "4E-1", "4F1", "4F-1", "4G1", "4G-1", "4H1", "4H-1", "8A1", "8A-1", "8B", "8C", "8D", "16A1", "16A-1", "16A3", "16A-3"]; chartbl_1024_dhg.OrderClassRepresentatives:= [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 16, 16, 16, 16]; chartbl_1024_dhg.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, 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*E(4), E(4), -1*E(4), -1*E(4), E(4), E(4), -1, E(4), -1*E(4), -1, -1, -1, -1, 1, 1, 1, E(4), E(4), -1*E(4), -1*E(4)], [1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, E(4), -1*E(4), E(4), E(4), -1*E(4), -1*E(4), -1, -1*E(4), E(4), -1, -1, -1, -1, 1, 1, 1, -1*E(4), -1*E(4), E(4), E(4)], [1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1*E(4), E(4), -1*E(4), -1*E(4), E(4), E(4), -1, E(4), -1*E(4), 1, 1, -1, -1, -1, 1, -1, -1*E(4), -1*E(4), E(4), E(4)], [1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, E(4), -1*E(4), E(4), E(4), -1*E(4), -1*E(4), -1, -1*E(4), E(4), 1, 1, -1, -1, -1, 1, -1, E(4), E(4), -1*E(4), -1*E(4)], [2, 2, 2, 2, 2, 2, -2, -2, -2, -2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 2, 0, -2, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, -2, 0, -2, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1*E(8)-E(8)^3, E(8)+E(8)^3, E(8)+E(8)^3, -1*E(8)-E(8)^3], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(8)+E(8)^3, -1*E(8)-E(8)^3, -1*E(8)-E(8)^3, E(8)+E(8)^3], [2, 2, 2, 2, 2, 2, 0, 0, 0, 0, -2, 2, -2, -2, -1-E(4), 1-E(4), -1-E(4), 1+E(4), -1+E(4), -1+E(4), 0, 1-E(4), 1+E(4), 0, 0, -2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 0, 0, 0, 0, -2, 2, -2, -2, -1+E(4), 1+E(4), -1+E(4), 1-E(4), -1-E(4), -1-E(4), 0, 1+E(4), 1-E(4), 0, 0, 2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 0, 0, 0, 0, -2, 2, -2, -2, 1-E(4), -1-E(4), 1-E(4), -1+E(4), 1+E(4), 1+E(4), 0, -1-E(4), -1+E(4), 0, 0, 2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 0, 0, 0, 0, -2, 2, -2, -2, 1+E(4), -1+E(4), 1+E(4), -1-E(4), 1-E(4), 1-E(4), 0, -1+E(4), -1-E(4), 0, 0, -2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, -2, -2, -2, -2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, E(8)+E(8)^-1], [2, 2, 2, 2, 2, 2, -2, -2, -2, -2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, -1*E(8)-E(8)^-1], [4, 4, 4, 4, 4, 4, 0, 0, 0, 0, -4, -4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, -4, 4, -4, 0, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0], [4, 4, 4, -4, 4, -4, 0, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 0, 0], [4, 4, 4, -4, 4, -4, 0, 0, 0, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, -4, 4, -4, 0, 0, 0, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 8, 8, 0, -8, 0, -4, 4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 8, 8, 0, -8, 0, 4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, -4, 0, 4, -4, 0, 0, 4, 0, 0, 0, 0, -2, 0, 2, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, -4, 0, 4, -4, 0, 0, 4, 0, 0, 0, 0, 2, 0, -2, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 4, 0, -4, 0, -4, 4, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 4, 0, -4, 0, -4, 4, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, -4, 0, 4, 4, 0, 0, -4, 0, 0, 0, 0, -2*E(4), 0, 2*E(4), 0, -2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, -4, 0, 4, 4, 0, 0, -4, 0, 0, 0, 0, 2*E(4), 0, -2*E(4), 0, 2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 4, 0, -4, 0, 4, -4, 0, 0, 0, 0, 0, 0, -2*E(4), 0, -2*E(4), 0, 0, 0, 2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 4, 0, -4, 0, 4, -4, 0, 0, 0, 0, 0, 0, 2*E(4), 0, 2*E(4), 0, 0, 0, -2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [16, 16, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; ConvertToLibraryCharacterTableNC(chartbl_1024_dhg);