# Group 64.186 downloaded from the LMFDB on 09 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(1180605043711004,64); a := GPC.1; b := GPC.2; c := GPC.3; GPerm := Group( (1,2)(3,7)(4,6)(5,13)(8,12)(9,11)(10,14)(15,16)(17,18), (2,6)(4,12)(5,11)(7,14)(8,9)(10,16)(13,15)(17,18), (1,3)(2,7)(4,10)(5,11)(6,14)(8,15)(9,13)(12,16)(17,18), (1,4,11,16,3,10,5,12)(2,8,13,14,7,15,9,6), (1,5,3,11)(2,9,7,13)(4,12,10,16)(6,15,14,8), (1,3)(2,7)(4,10)(5,11)(6,14)(8,15)(9,13)(12,16) ); GLZN := Group([[[ZmodnZObj(17,34), ZmodnZObj(18,34)], [ZmodnZObj(16,34), ZmodnZObj(17,34)]],[[ZmodnZObj(31,34), ZmodnZObj(14,34)], [ZmodnZObj(14,34), ZmodnZObj(3,34)]],[[ZmodnZObj(11,34), ZmodnZObj(4,34)], [ZmodnZObj(4,34), ZmodnZObj(23,34)]],[[ZmodnZObj(33,34), ZmodnZObj(0,34)], [ZmodnZObj(0,34), ZmodnZObj(33,34)]],[[ZmodnZObj(1,34), ZmodnZObj(17,34)], [ZmodnZObj(0,34), ZmodnZObj(1,34)]],[[ZmodnZObj(33,34), ZmodnZObj(0,34)], [ZmodnZObj(0,34), ZmodnZObj(1,34)]]]); GLZq := Group([[[ZmodnZObj(3,16), ZmodnZObj(3,16)], [ZmodnZObj(0,16), ZmodnZObj(3,16)]],[[ZmodnZObj(1,16), ZmodnZObj(2,16)], [ZmodnZObj(0,16), ZmodnZObj(1,16)]],[[ZmodnZObj(9,16), ZmodnZObj(0,16)], [ZmodnZObj(0,16), ZmodnZObj(9,16)]],[[ZmodnZObj(15,16), ZmodnZObj(12,16)], [ZmodnZObj(8,16), ZmodnZObj(9,16)]],[[ZmodnZObj(1,16), ZmodnZObj(4,16)], [ZmodnZObj(0,16), ZmodnZObj(1,16)]],[[ZmodnZObj(1,16), ZmodnZObj(8,16)], [ZmodnZObj(0,16), ZmodnZObj(1,16)]]]); # Booleans booleans_64_186 := 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_64_186:=rec(); chartbl_64_186.IsFinite:= true; chartbl_64_186.UnderlyingCharacteristic:= 0; chartbl_64_186.UnderlyingGroup:= GPC; chartbl_64_186.Size:= 64; chartbl_64_186.InfoText:= "Character table for group 64.186 downloaded from the LMFDB."; chartbl_64_186.Identifier:= " C2*D16 "; chartbl_64_186.NrConjugacyClasses:= 22; chartbl_64_186.ConjugacyClasses:= [ of ..., f1, f1*f6, f6, f2, f2*f3, f1*f2, f1*f2*f3, f1*f5, f5, f1*f4, f1*f4*f6, f4, f4*f5, f3, f3*f4, f3*f5, f3*f6, f1*f3, f1*f3*f4, f1*f3*f5, f1*f3*f6]; chartbl_64_186.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]; chartbl_64_186.ComputedPowerMaps:= [ , [1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 10, 10, 10, 10, 13, 14, 14, 13, 13, 14, 14, 13]]; chartbl_64_186.SizesCentralizers:= [64, 64, 64, 64, 8, 8, 8, 8, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32]; chartbl_64_186.ClassNames:= ["1A", "2A", "2B", "2C", "2D", "2E", "2F", "2G", "4A", "4B", "8A1", "8A3", "8B1", "8B3", "16A1", "16A3", "16A5", "16A7", "16B1", "16B3", "16B5", "16B7"]; chartbl_64_186.OrderClassRepresentatives:= [1, 2, 2, 2, 2, 2, 2, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16]; chartbl_64_186.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], [2, 2, -2, -2, 0, 0, 0, 0, -2, 2, 2, 2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 0, 0, 0, 0, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, -2, -2, 0, 0, 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, E(8)+E(8)^-1, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, -1*E(8)-E(8)^-1], [2, 2, -2, -2, 0, 0, 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, -1*E(8)-E(8)^-1, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, E(8)+E(8)^-1], [2, 2, 2, 2, 0, 0, 0, 0, -2, -2, 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, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, -1*E(8)-E(8)^-1], [2, 2, 2, 2, 0, 0, 0, 0, -2, -2, 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, -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, 0, 0, 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, -1*E(16)-E(16)^-1, E(16)^3+E(16)^-3, -1*E(16)-E(16)^-1, E(16)+E(16)^-1, -1*E(16)^3-E(16)^-3, E(16)+E(16)^-1, E(16)^3+E(16)^-3], [2, -2, -2, 2, 0, 0, 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, E(16)+E(16)^-1, -1*E(16)^3-E(16)^-3, E(16)+E(16)^-1, -1*E(16)-E(16)^-1, E(16)^3+E(16)^-3, -1*E(16)-E(16)^-1, -1*E(16)^3-E(16)^-3], [2, -2, -2, 2, 0, 0, 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, E(16)^3+E(16)^-3, 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)^3-E(16)^-3, E(16)+E(16)^-1], [2, -2, -2, 2, 0, 0, 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, -1*E(16)^3-E(16)^-3, -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)^3+E(16)^-3, -1*E(16)-E(16)^-1], [2, -2, 2, -2, 0, 0, 0, 0, 0, 0, -1*E(16)^2-E(16)^-2, 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, E(16)^3+E(16)^-3, -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, E(16)^3+E(16)^-3, E(16)+E(16)^-1], [2, -2, 2, -2, 0, 0, 0, 0, 0, 0, -1*E(16)^2-E(16)^-2, E(16)^2+E(16)^-2, -1*E(16)^2-E(16)^-2, E(16)^2+E(16)^-2, E(16)+E(16)^-1, -1*E(16)^3-E(16)^-3, E(16)+E(16)^-1, E(16)^3+E(16)^-3, E(16)^3+E(16)^-3, -1*E(16)-E(16)^-1, -1*E(16)^3-E(16)^-3, -1*E(16)-E(16)^-1], [2, -2, 2, -2, 0, 0, 0, 0, 0, 0, E(16)^2+E(16)^-2, -1*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, -1*E(16)-E(16)^-1, -1*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)-E(16)^-1, E(16)^3+E(16)^-3], [2, -2, 2, -2, 0, 0, 0, 0, 0, 0, E(16)^2+E(16)^-2, -1*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, E(16)+E(16)^-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)+E(16)^-1, -1*E(16)^3-E(16)^-3]]; ConvertToLibraryCharacterTableNC(chartbl_64_186);