# Group 256.24038 downloaded from the LMFDB on 14 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(1308869284194921156224087889467692,256); a := GPC.1; b := GPC.2; c := GPC.3; d := GPC.6; e := GPC.8; GPerm := Group( (1,2,7,9,5,10,6,11)(3,4,14,16,13,8,12,15)(17,18)(19,23)(20,21)(22,24), (1,3)(2,8)(4,10)(5,13)(6,12)(7,14)(9,15)(11,16)(18,21)(19,24), (2,9)(3,12)(6,7)(10,11)(13,14)(15,16), (1,4)(2,3)(5,8)(6,16)(7,15)(9,12)(10,13)(11,14)(17,19)(18,22)(20,24)(21,23), (1,5)(2,10)(3,13)(4,8)(6,7)(9,11)(12,14)(15,16)(17,20)(18,21)(19,24)(22,23), (1,6,5,7)(2,11,10,9)(3,12,13,14)(4,15,8,16), (1,6,5,7)(2,11,10,9)(3,14,13,12)(4,16,8,15), (1,5)(2,10)(3,13)(4,8)(6,7)(9,11)(12,14)(15,16) ); # Booleans booleans_256_24038 := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := true, metacyclic := false, monomial := true, nilpotent := true, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_256_24038:=rec(); chartbl_256_24038.IsFinite:= true; chartbl_256_24038.UnderlyingCharacteristic:= 0; chartbl_256_24038.UnderlyingGroup:= GPC; chartbl_256_24038.Size:= 256; chartbl_256_24038.InfoText:= "Character table for group 256.24038 downloaded from the LMFDB."; chartbl_256_24038.Identifier:= " C2^2.D4^2 "; chartbl_256_24038.NrConjugacyClasses:= 40; chartbl_256_24038.ConjugacyClasses:= [ of ..., f5, f5*f8, f8, f7*f8, f7, f2, f1, f2*f3, f1*f2, f1*f2*f8, f4, f4*f7, f4*f7*f8, f4*f8, f6, f6*f7, f6*f8, f5*f6, f2*f7, f1*f7, f2*f3*f7, f1*f3*f6, f1*f3*f6*f7, f1*f2*f6, f1*f2*f6*f8, f1*f2*f3, f1*f2*f3*f6, f3, f3*f5, f3*f7, f3*f7*f8, f3*f6, f3*f5*f6, f3*f6*f7, f3*f6*f7*f8, f2*f6, f1*f6, f1*f3, f2*f3*f6]; chartbl_256_24038.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, 35, 36, 37, 38, 39, 40]; chartbl_256_24038.ComputedPowerMaps:= [ , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 6, 6, 6, 6, 2, 2, 2, 3, 4, 3, 3, 4, 2, 12, 12, 12, 12, 13, 13, 13, 13, 14, 12, 15, 14]]; chartbl_256_24038.SizesCentralizers:= [256, 256, 256, 256, 128, 128, 32, 32, 32, 32, 32, 128, 128, 128, 128, 64, 64, 64, 64, 32, 32, 32, 32, 32, 32, 32, 16, 16, 64, 64, 64, 64, 64, 64, 64, 64, 16, 16, 16, 16]; chartbl_256_24038.ClassNames:= ["1A", "2A", "2B", "2C", "2D", "2E", "2F", "2G", "2H", "2I", "2J", "4A", "4B", "4C", "4D", "4E1", "4E-1", "4F1", "4F-1", "4G", "4H", "4I", "4J", "4K", "4L1", "4L-1", "4M", "4N", "8A1", "8A3", "8B1", "8B-1", "8C1", "8C-1", "8D1", "8D-1", "8E", "8F", "8G", "8H"]; chartbl_256_24038.OrderClassRepresentatives:= [1, 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, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8]; chartbl_256_24038.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, 1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1], [1, 1, 1, 1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1], [1, 1, 1, 1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1], [1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, 1], [1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1], [1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1], [1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1], [1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1], [1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -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, 2, -2, -2, 0, 0, -2, 0, -2, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, -2, 0, -2, 0, 2, 0, 2, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, -2, 0, 2, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 2, 0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 0, -2, 0, 2, -2, -2, 2, 0, 0, 0, 0, -2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, -2, 0, 2, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 0, -2, 0, 2, -2, -2, 2, 0, 0, 0, 0, 2, 2, 0, -2, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 2, 0, -2, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 0, 2, 0, 2, -2, -2, 2, 0, 0, 0, 0, -2, -2, 0, 2, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 2, 0, -2, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 0, 2, 0, 2, -2, -2, 2, 0, 0, 0, 0, 2, -2, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, -2, 0, 2, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 2, 0, -2, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 2, 0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 2, 0, 2, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0, 0, -2, 0, -2, 0, 2, 0, 2, 0, 0, 0, 0, 0], [2, -2, 2, -2, 2, -2, -2, 2, 0, 0, 0, -2, 2, -2, 2, -2*E(4), 2*E(4), -2*E(4), 2*E(4), 0, 0, -2*E(4), 0, 0, 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, 2, -2, 2, -2, -2, 2, 0, 0, 0, -2, 2, -2, 2, 2*E(4), -2*E(4), 2*E(4), -2*E(4), 0, 0, 2*E(4), 0, 0, -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, 2, -2, 2, -2, 2, -2, 0, 0, 0, -2, 2, -2, 2, -2*E(4), 2*E(4), -2*E(4), 2*E(4), 0, 0, 2*E(4), 0, 0, -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, 2, -2, 2, -2, 2, -2, 0, 0, 0, -2, 2, -2, 2, 2*E(4), -2*E(4), 2*E(4), -2*E(4), 0, 0, -2*E(4), 0, 0, 2*E(4), 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, -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, 0], [4, -4, 4, -4, -4, 4, 0, 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, 0, 0, 0, 0, 0], [4, -4, 4, -4, 4, -4, 0, 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, 0, 0, 0, 0, 0], [4, 4, 4, 4, 4, 4, 0, 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, 0, 0, 0, 0, 0], [4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(8)-2*E(8)^3, -2*E(8)-2*E(8)^3, 2*E(8)+2*E(8)^3, 0, 0, 0, 0, 2*E(8)+2*E(8)^3, 0, 0, 0, 0], [4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(8)+2*E(8)^3, 2*E(8)+2*E(8)^3, -2*E(8)-2*E(8)^3, 0, 0, 0, 0, -2*E(8)-2*E(8)^3, 0, 0, 0, 0], [4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(8)-2*E(8)^3, 2*E(8)+2*E(8)^3, 2*E(8)+2*E(8)^3, 0, 0, 0, 0, -2*E(8)-2*E(8)^3, 0, 0, 0, 0], [4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(8)+2*E(8)^3, -2*E(8)-2*E(8)^3, -2*E(8)-2*E(8)^3, 0, 0, 0, 0, 2*E(8)+2*E(8)^3, 0, 0, 0, 0], [4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(8)^2, -2*E(8)^2, 2*E(8)^2, 2*E(8)^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(8)-2*E(8)^3, 2*E(8)+2*E(8)^-1, 2*E(8)+2*E(8)^3, -2*E(8)-2*E(8)^-1, 0, 0, 0, 0, 0], [4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(8)^2, 2*E(8)^2, -2*E(8)^2, -2*E(8)^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(8)+2*E(8)^3, 2*E(8)+2*E(8)^-1, -2*E(8)-2*E(8)^3, -2*E(8)-2*E(8)^-1, 0, 0, 0, 0, 0], [4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(8)^2, -2*E(8)^2, 2*E(8)^2, 2*E(8)^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(8)+2*E(8)^3, -2*E(8)-2*E(8)^-1, -2*E(8)-2*E(8)^3, 2*E(8)+2*E(8)^-1, 0, 0, 0, 0, 0], [4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(8)^2, 2*E(8)^2, -2*E(8)^2, -2*E(8)^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(8)-2*E(8)^3, -2*E(8)-2*E(8)^-1, 2*E(8)+2*E(8)^3, 2*E(8)+2*E(8)^-1, 0, 0, 0, 0, 0]]; ConvertToLibraryCharacterTableNC(chartbl_256_24038);