# Group 1024.dhr downloaded from the LMFDB on 15 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(243071448734599231902892238864320355685685536137397247573039650439739210433045704071,1024); a := GPC.1; b := GPC.3; c := GPC.5; d := GPC.6; e := GPC.7; f := GPC.8; GPerm := Group( (1,9,8,15,3,12,5,13)(2,10,7,16,4,11,6,14), (3,4)(5,6)(7,8)(13,16,14,15) ); # Booleans booleans_1024_dhr := 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_dhr:=rec(); chartbl_1024_dhr.IsFinite:= true; chartbl_1024_dhr.UnderlyingCharacteristic:= 0; chartbl_1024_dhr.UnderlyingGroup:= GPerm; chartbl_1024_dhr.Size:= 1024; chartbl_1024_dhr.InfoText:= "Character table for group 1024.dhr downloaded from the LMFDB."; chartbl_1024_dhr.Identifier:= " C2^4.C2wrC4 "; chartbl_1024_dhr.NrConjugacyClasses:= 34; chartbl_1024_dhr.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), (5,6)(7,8)(13,14)(15,16), (5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (9,10)(11,12), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15), (1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16), (1,6)(2,5)(3,8)(4,7)(9,14)(10,13)(11,15)(12,16), (1,2)(7,8)(9,12,10,11)(13,16,14,15), (1,2)(7,8)(9,11,10,12)(13,15,14,16), (7,8)(9,11,10,12), (7,8)(9,11,10,12)(13,14)(15,16), (5,7,6,8)(9,10)(11,12)(15,16), (3,4)(7,8)(9,11,10,12)(13,15,14,16), (3,4)(5,6)(7,8)(9,10)(11,12)(13,15,14,16), (1,6,2,5)(3,7,4,8)(9,16,10,15)(11,14,12,13), (5,7)(6,8)(9,11,10,12)(15,16), (3,4)(5,7,6,8)(9,10)(11,12)(13,15)(14,16), (1,5)(2,6)(3,7)(4,8)(9,13,10,14)(11,16,12,15), (3,4)(5,7)(6,8)(9,11,10,12)(13,15)(14,16), (3,4)(5,7)(6,8)(9,12,10,11)(13,15)(14,16), (1,6,4,7)(2,5,3,8)(9,15,11,13)(10,16,12,14), (1,7,4,6)(2,8,3,5)(9,13,11,15)(10,14,12,16), (1,15,6,11)(2,16,5,12)(3,13,8,10)(4,14,7,9), (1,11,6,15)(2,12,5,16)(3,10,8,13)(4,9,7,14), (1,8,2,7)(3,6)(4,5)(9,14,12,15,10,13,11,16), (1,7,2,8)(3,6)(4,5)(9,16,11,13,10,15,12,14), (1,9,6,16,2,10,5,15)(3,11,7,14,4,12,8,13), (1,15,5,10,2,16,6,9)(3,13,8,12,4,14,7,11), (1,11,6,13,4,9,7,15)(2,12,5,14,3,10,8,16), (1,15,7,9,4,13,6,11)(2,16,8,10,3,14,5,12), (1,13,7,11,4,15,6,9)(2,14,8,12,3,16,5,10), (1,9,6,15,4,11,7,13)(2,10,5,16,3,12,8,14)]; chartbl_1024_dhr.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_dhr.ComputedPowerMaps:= [ , [1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 6, 6, 6, 3, 6, 2, 6, 6, 3, 6, 6, 8, 8, 9, 9, 10, 11, 17, 17, 23, 24, 24, 23]]; chartbl_1024_dhr.SizesCentralizers:= [1024, 1024, 512, 256, 256, 256, 128, 128, 64, 128, 128, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, 32, 32, 32, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16]; chartbl_1024_dhr.ClassNames:= ["1A", "2A", "2B", "2C", "2D", "2E", "2F", "2G", "2H", "4A1", "4A-1", "4B", "4C", "4D", "4E", "4F", "4G", "4H", "4I", "4J", "4K1", "4K-1", "4L1", "4L-1", "4M1", "4M-1", "8A1", "8A-1", "8B1", "8B-1", "8C1", "8C-1", "8C3", "8C-3"]; chartbl_1024_dhr.OrderClassRepresentatives:= [1, 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, 8, 8, 8, 8, 8, 8, 8, 8]; chartbl_1024_dhr.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, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1*E(4), E(4), E(4), 1, 1, -1*E(4), -1*E(4), E(4), E(4), -1*E(4)], [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), -1*E(4), -1*E(4), 1, 1, E(4), E(4), -1*E(4), -1*E(4), E(4)], [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), E(4), -1, -1, -1*E(4), E(4), -1*E(4), -1*E(4), E(4)], [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), -1*E(4), -1*E(4), -1, -1, E(4), -1*E(4), E(4), E(4), -1*E(4)], [2, 2, 2, 2, 2, 2, 2, 2, -2, 2, 2, 0, 0, 2, 0, -2, 0, 0, 0, 2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 2, 0, 0, 0, -2, -2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 2, -2, 2, -2, -2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -1*E(8)-E(8)^3, -1*E(8)-E(8)^3, E(8)+E(8)^3, E(8)+E(8)^3], [2, 2, 2, 2, -2, -2, 2, -2, 2, -2, -2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, E(8)+E(8)^3, E(8)+E(8)^3, -1*E(8)-E(8)^3, -1*E(8)-E(8)^3], [2, 2, 2, 2, -2, -2, 2, -2, 0, 2, 2, 0, 0, -2, 0, 0, 0, 0, 0, -2*E(4), 2*E(4), 0, 0, 0, -1+E(4), -1-E(4), 1+E(4), 0, 0, 1-E(4), 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 2, -2, 0, 2, 2, 0, 0, -2, 0, 0, 0, 0, 0, 2*E(4), -2*E(4), 0, 0, 0, -1-E(4), -1+E(4), 1-E(4), 0, 0, 1+E(4), 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 2, -2, 0, 2, 2, 0, 0, -2, 0, 0, 0, 0, 0, -2*E(4), 2*E(4), 0, 0, 0, 1-E(4), 1+E(4), -1-E(4), 0, 0, -1+E(4), 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 2, -2, 0, 2, 2, 0, 0, -2, 0, 0, 0, 0, 0, 2*E(4), -2*E(4), 0, 0, 0, 1+E(4), 1-E(4), -1+E(4), 0, 0, -1-E(4), 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 2, -2, -2, -2, -2, 0, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, 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, 0, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, 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, 4, 4, 0, -4, -4, 0, 0, -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, -4, -4, 0, 0, 0, -2, -2, 0, -2, 0, -2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, 4, 4, 4, -4, -4, 0, 0, 0, 2, 2, 0, 2, 0, 2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, 4, -4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(4), 2*E(4), 0, 0, 0, 0, 0], [4, 4, 4, 4, -4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(4), -2*E(4), 0, 0, 0, 0, 0], [8, 8, 8, -8, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 8, 8, -8, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 0, -4, 4, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, -4, 0, 4, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 0, -4, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -4, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 8, -8, 0, 0, 0, 0, 0, 0, -4*E(4), 4*E(4), -2, 2, 0, 2, 0, -2, 2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 8, -8, 0, 0, 0, 0, 0, 0, 4*E(4), -4*E(4), -2, 2, 0, 2, 0, -2, -2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 8, -8, 0, 0, 0, 0, 0, 0, -4*E(4), 4*E(4), 2, -2, 0, -2, 0, 2, -2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 8, -8, 0, 0, 0, 0, 0, 0, 4*E(4), -4*E(4), 2, -2, 0, -2, 0, 2, 2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [16, -16, 0, 0, 8, -8, 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_dhr);