# Group 192.775 downloaded from the LMFDB on 05 November 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(166993391821721322595603969,192); a := GPC.1; b := GPC.2; c := GPC.3; d := GPC.6; GPerm := Group( (1,2,3,6,4,5,7,8)(10,11), (12,13), (2,5)(3,7), (1,3,4,7)(2,6,5,8), (2,5)(6,8), (1,4)(2,5)(3,7)(6,8), (9,10,11) ); GLZ := Group([[[-1, 0, 1, 1, 0, 0], [-1, 0, 1, 0, 0, 0], [0, -1, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, -1, 1]], [[-1, 0, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 0], [-1, 0, 1, 1, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]], [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, -1, 1]]]); GLZN := Group([[[ZmodnZObj(1,48), ZmodnZObj(16,48)], [ZmodnZObj(0,48), ZmodnZObj(1,48)]],[[ZmodnZObj(1,48), ZmodnZObj(24,48)], [ZmodnZObj(0,48), ZmodnZObj(1,48)]],[[ZmodnZObj(31,48), ZmodnZObj(6,48)], [ZmodnZObj(0,48), ZmodnZObj(1,48)]],[[ZmodnZObj(25,48), ZmodnZObj(24,48)], [ZmodnZObj(24,48), ZmodnZObj(25,48)]],[[ZmodnZObj(43,48), ZmodnZObj(24,48)], [ZmodnZObj(12,48), ZmodnZObj(19,48)]],[[ZmodnZObj(31,48), ZmodnZObj(20,48)], [ZmodnZObj(9,48), ZmodnZObj(5,48)]],[[ZmodnZObj(13,48), ZmodnZObj(0,48)], [ZmodnZObj(12,48), ZmodnZObj(37,48)]]]); # Booleans booleans_192_775 := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := true, metacyclic := false, monomial := true, nilpotent := false, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_192_775:=rec(); chartbl_192_775.IsFinite:= true; chartbl_192_775.UnderlyingCharacteristic:= 0; chartbl_192_775.UnderlyingGroup:= GPC; chartbl_192_775.Size:= 192; chartbl_192_775.InfoText:= "Character table for group 192.775 downloaded from the LMFDB."; chartbl_192_775.Identifier:= " C2*C12.D4 "; chartbl_192_775.NrConjugacyClasses:= 42; chartbl_192_775.ConjugacyClasses:= [ of ..., f6*f7, f5*f6*f7, f5, f2, f2*f6*f7, f1, f1*f6*f7, f1*f4, f1*f4*f6*f7, f7, f4, f2*f4*f5, f4*f5*f6*f7, f2*f4*f5*f6*f7, f6, f5*f6, f5*f7^2, f2*f7, f2*f7^2, f2*f6, f2*f5*f6, f1*f7, f1*f7^2, f1*f6, f1*f2*f6, f1*f4*f7, f1*f4*f7^2, f1*f4*f6, f1*f2*f4*f6, f3, f3*f4, f3*f6, f3*f4*f6, f1*f3, f1*f3*f4, f1*f3*f6, f1*f3*f4*f6, f4*f7, f4*f6, f2*f4*f7, f2*f4*f6]; chartbl_192_775.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, 41, 42]; chartbl_192_775.ComputedPowerMaps:= [ , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 4, 4, 4, 4, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 18, 18, 18, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 13, 14, 15, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 32, 31, 34, 33, 36, 35, 38, 37, 12, 14, 13, 15]]; chartbl_192_775.SizesCentralizers:= [192, 192, 192, 192, 96, 96, 48, 48, 48, 48, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 48, 48, 48, 48, 48, 48, 48, 48, 16, 16, 16, 16, 16, 16, 16, 16, 48, 48, 48, 48]; chartbl_192_775.ClassNames:= ["1A", "2A", "2B", "2C", "2D", "2E", "2F", "2G", "2H", "2I", "3A", "4A", "4B", "4C", "4D", "6A", "6B", "6C", "6D1", "6D-1", "6E1", "6E-1", "6F1", "6F-1", "6G1", "6G-1", "6H1", "6H-1", "6I1", "6I-1", "8A1", "8A-1", "8B1", "8B-1", "8C1", "8C-1", "8D1", "8D-1", "12A", "12B", "12C", "12D"]; chartbl_192_775.OrderClassRepresentatives:= [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 12, 12, 12, 12]; chartbl_192_775.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*E(4), E(4), E(4), -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, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, E(4), -1*E(4), -1*E(4), 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, -1, 1, 1, 1, 1, -1, -1, -1, -1, -1*E(4), E(4), E(4), -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, -1, 1, 1, 1, 1, -1, -1, -1, -1, E(4), -1*E(4), -1*E(4), 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, 1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1*E(4), E(4), -1*E(4), E(4), -1*E(4), 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, 1, 1, -1, 1, -1, -1, 1, 1, -1, E(4), -1*E(4), E(4), -1*E(4), E(4), -1*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, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1*E(4), E(4), -1*E(4), E(4), E(4), -1*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, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, E(4), -1*E(4), E(4), -1*E(4), -1*E(4), E(4), -1*E(4), E(4), -1, -1, -1, -1], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -1, 2, 2, 2, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1], [2, 2, -2, -2, -2, 2, 0, 0, 0, 0, 2, -2, -2, 2, 2, 2, 2, -2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -2, -2], [2, 2, -2, -2, -2, 2, 0, 0, 0, 0, 2, 2, 2, -2, -2, 2, 2, -2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 2, 2], [2, 2, 2, 2, -2, -2, 0, 0, 0, 0, 2, -2, 2, 2, -2, -2, -2, -2, 2, 2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, -2, 2], [2, 2, 2, 2, -2, -2, 0, 0, 0, 0, 2, 2, -2, -2, 2, -2, -2, -2, 2, 2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 2, -2], [2, 2, -2, -2, 2, -2, -2, 2, -2, 2, -1, -2, 2, -2, 2, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, -1], [2, 2, -2, -2, 2, -2, 2, -2, 2, -2, -1, -2, 2, -2, 2, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, -1], [2, 2, 2, 2, 2, 2, -2, -2, -2, -2, -1, 2, 2, 2, 2, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1], [2, 2, -2, -2, 2, -2, -2, 2, 2, -2, -1, 2, -2, 2, -2, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1], [2, 2, -2, -2, 2, -2, 2, -2, -2, 2, -1, 2, -2, 2, -2, 1, 1, -1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1], [2, 2, 2, 2, 2, 2, -2, -2, 2, 2, -1, -2, -2, -2, -2, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2, 2, 2, -2, -2, -1, -2, -2, -2, -2, -1, -1, -1, -1, -1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [2, 2, -2, -2, -2, 2, 0, 0, 0, 0, -1, -2, -2, 2, 2, -1, -1, 1, -1, 1, 1, 1, -1-2*E(3), 1+2*E(3), 1+2*E(3), -1-2*E(3), 1+2*E(3), 1+2*E(3), -1-2*E(3), -1-2*E(3), 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 1], [2, 2, -2, -2, -2, 2, 0, 0, 0, 0, -1, -2, -2, 2, 2, -1, -1, 1, -1, 1, 1, 1, 1+2*E(3), -1-2*E(3), -1-2*E(3), 1+2*E(3), -1-2*E(3), -1-2*E(3), 1+2*E(3), 1+2*E(3), 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 1], [2, 2, -2, -2, -2, 2, 0, 0, 0, 0, -1, 2, 2, -2, -2, -1, -1, 1, -1, 1, 1, 1, -1-2*E(3), -1-2*E(3), 1+2*E(3), 1+2*E(3), -1-2*E(3), 1+2*E(3), -1-2*E(3), 1+2*E(3), 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1], [2, 2, -2, -2, -2, 2, 0, 0, 0, 0, -1, 2, 2, -2, -2, -1, -1, 1, -1, 1, 1, 1, 1+2*E(3), 1+2*E(3), -1-2*E(3), -1-2*E(3), 1+2*E(3), -1-2*E(3), 1+2*E(3), -1-2*E(3), 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1], [2, 2, 2, 2, -2, -2, 0, 0, 0, 0, -1, -2, 2, 2, -2, 1, 1, 1, -1, -1, 1, -1, -1-2*E(3), 1+2*E(3), 1+2*E(3), -1-2*E(3), -1-2*E(3), -1-2*E(3), 1+2*E(3), 1+2*E(3), 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1], [2, 2, 2, 2, -2, -2, 0, 0, 0, 0, -1, -2, 2, 2, -2, 1, 1, 1, -1, -1, 1, -1, 1+2*E(3), -1-2*E(3), -1-2*E(3), 1+2*E(3), 1+2*E(3), 1+2*E(3), -1-2*E(3), -1-2*E(3), 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1], [2, 2, 2, 2, -2, -2, 0, 0, 0, 0, -1, 2, -2, -2, 2, 1, 1, 1, -1, -1, 1, -1, -1-2*E(3), -1-2*E(3), 1+2*E(3), 1+2*E(3), 1+2*E(3), -1-2*E(3), 1+2*E(3), -1-2*E(3), 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1], [2, 2, 2, 2, -2, -2, 0, 0, 0, 0, -1, 2, -2, -2, 2, 1, 1, 1, -1, -1, 1, -1, 1+2*E(3), 1+2*E(3), -1-2*E(3), -1-2*E(3), -1-2*E(3), 1+2*E(3), -1-2*E(3), 1+2*E(3), 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1], [4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -4, -4, 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, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -4, 4, 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, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2-4*E(3), 2+4*E(3), -2-4*E(3), 2, 2, 2+4*E(3), -2, 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, -2, 0, 0, 0, 0, 2+4*E(3), -2-4*E(3), 2+4*E(3), 2, 2, -2-4*E(3), -2, 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, -2, 0, 0, 0, 0, -2-4*E(3), 2+4*E(3), 2+4*E(3), 2, -2, -2-4*E(3), 2, 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, -2, 0, 0, 0, 0, 2+4*E(3), -2-4*E(3), -2-4*E(3), 2, -2, 2+4*E(3), 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; ConvertToLibraryCharacterTableNC(chartbl_192_775);