# Group 96.6 downloaded from the LMFDB on 18 September 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(494470908942143314788225,96); a := GPC.1; b := GPC.2; GPerm := Group( (1,2)(3,4)(5,15)(6,9)(7,10)(8,13)(11,16)(12,14)(18,19), (1,3,10,8,5,11,14,9,6,12,16,15,13,7,4,2), (1,4,13,16,6,14,5,10)(2,7,15,12,9,11,8,3), (1,5,6,13)(2,8,9,15)(3,11,12,7)(4,10,14,16), (1,6)(2,9)(3,12)(4,14)(5,13)(7,11)(8,15)(10,16), (17,18,19) ); GLFp := Group([[[ Z(47)^19, Z(47) ], [ Z(47)^0, Z(47)^19 ]], [[ Z(47)^0, 0*Z(47) ], [ 0*Z(47), Z(47)^23 ]]]); # Booleans booleans_96_6 := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := true, metacyclic := true, monomial := true, nilpotent := false, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_96_6:=rec(); chartbl_96_6.IsFinite:= true; chartbl_96_6.UnderlyingCharacteristic:= 0; chartbl_96_6.UnderlyingGroup:= GPC; chartbl_96_6.Size:= 96; chartbl_96_6.InfoText:= "Character table for group 96.6 downloaded from the LMFDB."; chartbl_96_6.Identifier:= " D48 "; chartbl_96_6.NrConjugacyClasses:= 27; chartbl_96_6.ConjugacyClasses:= [ of ..., f5*f6, f1*f5*f6, f1*f2*f3*f5*f6, f6, f4*f5, f5, f3*f4, f3*f6, f4, f4*f6, f2*f3, f2*f5, f2*f3*f4*f5, f2*f4*f6, f3, f3*f5, f3*f4*f5, f3*f4*f6, f2, f2*f4, f2*f3*f4, f2*f3*f5, f2*f4*f5, f2*f6, f2*f3*f6, f2*f5*f6]; chartbl_96_6.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]; chartbl_96_6.ComputedPowerMaps:= [ , [1, 1, 1, 1, 5, 2, 5, 6, 6, 7, 7, 8, 9, 9, 8, 10, 11, 11, 10, 16, 17, 18, 19, 19, 18, 17, 16], [1, 2, 3, 4, 1, 6, 2, 9, 8, 6, 6, 13, 15, 12, 14, 8, 9, 8, 9, 12, 14, 15, 14, 13, 12, 13, 15]]; chartbl_96_6.SizesCentralizers:= [96, 96, 4, 4, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48]; chartbl_96_6.ClassNames:= ["1A", "2A", "2B", "2C", "3A", "4A", "6A", "8A1", "8A3", "12A1", "12A5", "16A1", "16A3", "16A5", "16A7", "24A1", "24A5", "24A7", "24A11", "48A1", "48A5", "48A7", "48A11", "48A13", "48A17", "48A19", "48A23"]; chartbl_96_6.OrderClassRepresentatives:= [1, 2, 2, 2, 3, 4, 6, 8, 8, 12, 12, 16, 16, 16, 16, 24, 24, 24, 24, 48, 48, 48, 48, 48, 48, 48, 48]; chartbl_96_6.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], [2, 2, 0, 0, -1, 2, -1, 2, 2, -1, -1, 2, 2, 2, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [2, 2, 0, 0, 2, 2, 2, -2, -2, 2, 2, 0, 0, 0, 0, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 0, 0, -1, 2, -1, 2, 2, -1, -1, -2, -2, -2, -2, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1], [2, 2, 0, 0, 2, -2, 2, 0, 0, -2, -2, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, 0, 0, 0, 0, E(8)+E(8)^-1, -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, E(8)+E(8)^-1, -1*E(8)-E(8)^-1], [2, 2, 0, 0, 2, -2, 2, 0, 0, -2, -2, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, 0, 0, 0, 0, -1*E(8)-E(8)^-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, -1*E(8)-E(8)^-1, E(8)+E(8)^-1], [2, 2, 0, 0, -1, 2, -1, -2, -2, -1, -1, 0, 0, 0, 0, 1, 1, 1, 1, -1*E(12)-E(12)^-1, E(12)+E(12)^-1, -1*E(12)-E(12)^-1, E(12)+E(12)^-1, -1*E(12)-E(12)^-1, E(12)+E(12)^-1, E(12)+E(12)^-1, -1*E(12)-E(12)^-1], [2, 2, 0, 0, -1, 2, -1, -2, -2, -1, -1, 0, 0, 0, 0, 1, 1, 1, 1, E(12)+E(12)^-1, -1*E(12)-E(12)^-1, E(12)+E(12)^-1, -1*E(12)-E(12)^-1, E(12)+E(12)^-1, -1*E(12)-E(12)^-1, -1*E(12)-E(12)^-1, E(12)+E(12)^-1], [2, -2, 0, 0, 2, 0, -2, -1*E(16)^2-E(16)^-2, E(16)^2+E(16)^-2, 0, 0, -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)^2-E(16)^-2, -1*E(16)^2-E(16)^-2, 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)^3-E(16)^-3, -1*E(16)^3-E(16)^-3, E(16)+E(16)^-1, -1*E(16)-E(16)^-1, E(16)+E(16)^-1, E(16)^3+E(16)^-3], [2, -2, 0, 0, 2, 0, -2, -1*E(16)^2-E(16)^-2, E(16)^2+E(16)^-2, 0, 0, E(16)^3+E(16)^-3, -1*E(16)-E(16)^-1, E(16)+E(16)^-1, -1*E(16)^3-E(16)^-3, -1*E(16)^2-E(16)^-2, -1*E(16)^2-E(16)^-2, 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)^3+E(16)^-3, E(16)^3+E(16)^-3, -1*E(16)-E(16)^-1, E(16)+E(16)^-1, -1*E(16)-E(16)^-1, -1*E(16)^3-E(16)^-3], [2, -2, 0, 0, 2, 0, -2, E(16)^2+E(16)^-2, -1*E(16)^2-E(16)^-2, 0, 0, -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)^2+E(16)^-2, E(16)^2+E(16)^-2, -1*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)-E(16)^-1, -1*E(16)-E(16)^-1, -1*E(16)^3-E(16)^-3, E(16)^3+E(16)^-3, -1*E(16)^3-E(16)^-3, E(16)+E(16)^-1], [2, -2, 0, 0, 2, 0, -2, E(16)^2+E(16)^-2, -1*E(16)^2-E(16)^-2, 0, 0, E(16)+E(16)^-1, E(16)^3+E(16)^-3, -1*E(16)^3-E(16)^-3, -1*E(16)-E(16)^-1, E(16)^2+E(16)^-2, E(16)^2+E(16)^-2, -1*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)+E(16)^-1, E(16)+E(16)^-1, E(16)^3+E(16)^-3, -1*E(16)^3-E(16)^-3, E(16)^3+E(16)^-3, -1*E(16)-E(16)^-1], [2, 2, 0, 0, -1, -2, -1, 0, 0, 1, 1, -1*E(24)^3-E(24)^-3, E(24)^3+E(24)^-3, E(24)^3+E(24)^-3, -1*E(24)^3-E(24)^-3, -1*E(24)^2-E(24)^-2, E(24)^2+E(24)^-2, -1*E(24)^2-E(24)^-2, E(24)^2+E(24)^-2, -1*E(24)-E(24)^-1, -1*E(24)^5-E(24)^-5, E(24)+E(24)^-1, -1*E(24)^5-E(24)^-5, -1*E(24)-E(24)^-1, E(24)^5+E(24)^-5, E(24)^5+E(24)^-5, E(24)+E(24)^-1], [2, 2, 0, 0, -1, -2, -1, 0, 0, 1, 1, -1*E(24)^3-E(24)^-3, E(24)^3+E(24)^-3, E(24)^3+E(24)^-3, -1*E(24)^3-E(24)^-3, E(24)^2+E(24)^-2, -1*E(24)^2-E(24)^-2, E(24)^2+E(24)^-2, -1*E(24)^2-E(24)^-2, E(24)^5+E(24)^-5, E(24)+E(24)^-1, -1*E(24)^5-E(24)^-5, E(24)+E(24)^-1, E(24)^5+E(24)^-5, -1*E(24)-E(24)^-1, -1*E(24)-E(24)^-1, -1*E(24)^5-E(24)^-5], [2, 2, 0, 0, -1, -2, -1, 0, 0, 1, 1, E(24)^3+E(24)^-3, -1*E(24)^3-E(24)^-3, -1*E(24)^3-E(24)^-3, E(24)^3+E(24)^-3, -1*E(24)^2-E(24)^-2, E(24)^2+E(24)^-2, -1*E(24)^2-E(24)^-2, E(24)^2+E(24)^-2, E(24)+E(24)^-1, E(24)^5+E(24)^-5, -1*E(24)-E(24)^-1, E(24)^5+E(24)^-5, E(24)+E(24)^-1, -1*E(24)^5-E(24)^-5, -1*E(24)^5-E(24)^-5, -1*E(24)-E(24)^-1], [2, 2, 0, 0, -1, -2, -1, 0, 0, 1, 1, E(24)^3+E(24)^-3, -1*E(24)^3-E(24)^-3, -1*E(24)^3-E(24)^-3, E(24)^3+E(24)^-3, E(24)^2+E(24)^-2, -1*E(24)^2-E(24)^-2, E(24)^2+E(24)^-2, -1*E(24)^2-E(24)^-2, -1*E(24)^5-E(24)^-5, -1*E(24)-E(24)^-1, E(24)^5+E(24)^-5, -1*E(24)-E(24)^-1, -1*E(24)^5-E(24)^-5, E(24)+E(24)^-1, E(24)+E(24)^-1, E(24)^5+E(24)^-5], [2, -2, 0, 0, -1, 0, 1, -1*E(48)^6-E(48)^-6, E(48)^6+E(48)^-6, -1*E(48)^4-E(48)^-4, E(48)^4+E(48)^-4, -1*E(48)^9-E(48)^-9, E(48)^3+E(48)^-3, -1*E(48)^3-E(48)^-3, E(48)^9+E(48)^-9, E(48)^2+E(48)^-2, -1*E(48)^10-E(48)^-10, -1*E(48)^2-E(48)^-2, E(48)^10+E(48)^-10, E(48)^5+E(48)^-5, -1*E(48)-E(48)^-1, -1*E(48)^7-E(48)^-7, E(48)+E(48)^-1, -1*E(48)^5-E(48)^-5, E(48)^11+E(48)^-11, -1*E(48)^11-E(48)^-11, E(48)^7+E(48)^-7], [2, -2, 0, 0, -1, 0, 1, -1*E(48)^6-E(48)^-6, E(48)^6+E(48)^-6, -1*E(48)^4-E(48)^-4, E(48)^4+E(48)^-4, E(48)^9+E(48)^-9, -1*E(48)^3-E(48)^-3, E(48)^3+E(48)^-3, -1*E(48)^9-E(48)^-9, E(48)^2+E(48)^-2, -1*E(48)^10-E(48)^-10, -1*E(48)^2-E(48)^-2, E(48)^10+E(48)^-10, -1*E(48)^5-E(48)^-5, E(48)+E(48)^-1, E(48)^7+E(48)^-7, -1*E(48)-E(48)^-1, E(48)^5+E(48)^-5, -1*E(48)^11-E(48)^-11, E(48)^11+E(48)^-11, -1*E(48)^7-E(48)^-7], [2, -2, 0, 0, -1, 0, 1, -1*E(48)^6-E(48)^-6, E(48)^6+E(48)^-6, E(48)^4+E(48)^-4, -1*E(48)^4-E(48)^-4, -1*E(48)^9-E(48)^-9, E(48)^3+E(48)^-3, -1*E(48)^3-E(48)^-3, E(48)^9+E(48)^-9, -1*E(48)^10-E(48)^-10, E(48)^2+E(48)^-2, E(48)^10+E(48)^-10, -1*E(48)^2-E(48)^-2, E(48)^11+E(48)^-11, E(48)^7+E(48)^-7, E(48)+E(48)^-1, -1*E(48)^7-E(48)^-7, -1*E(48)^11-E(48)^-11, E(48)^5+E(48)^-5, -1*E(48)^5-E(48)^-5, -1*E(48)-E(48)^-1], [2, -2, 0, 0, -1, 0, 1, -1*E(48)^6-E(48)^-6, E(48)^6+E(48)^-6, E(48)^4+E(48)^-4, -1*E(48)^4-E(48)^-4, E(48)^9+E(48)^-9, -1*E(48)^3-E(48)^-3, E(48)^3+E(48)^-3, -1*E(48)^9-E(48)^-9, -1*E(48)^10-E(48)^-10, E(48)^2+E(48)^-2, E(48)^10+E(48)^-10, -1*E(48)^2-E(48)^-2, -1*E(48)^11-E(48)^-11, -1*E(48)^7-E(48)^-7, -1*E(48)-E(48)^-1, E(48)^7+E(48)^-7, E(48)^11+E(48)^-11, -1*E(48)^5-E(48)^-5, E(48)^5+E(48)^-5, E(48)+E(48)^-1], [2, -2, 0, 0, -1, 0, 1, E(48)^6+E(48)^-6, -1*E(48)^6-E(48)^-6, -1*E(48)^4-E(48)^-4, E(48)^4+E(48)^-4, -1*E(48)^3-E(48)^-3, -1*E(48)^9-E(48)^-9, E(48)^9+E(48)^-9, E(48)^3+E(48)^-3, -1*E(48)^2-E(48)^-2, E(48)^10+E(48)^-10, E(48)^2+E(48)^-2, -1*E(48)^10-E(48)^-10, E(48)^7+E(48)^-7, -1*E(48)^11-E(48)^-11, E(48)^5+E(48)^-5, E(48)^11+E(48)^-11, -1*E(48)^7-E(48)^-7, -1*E(48)-E(48)^-1, E(48)+E(48)^-1, -1*E(48)^5-E(48)^-5], [2, -2, 0, 0, -1, 0, 1, E(48)^6+E(48)^-6, -1*E(48)^6-E(48)^-6, -1*E(48)^4-E(48)^-4, E(48)^4+E(48)^-4, E(48)^3+E(48)^-3, E(48)^9+E(48)^-9, -1*E(48)^9-E(48)^-9, -1*E(48)^3-E(48)^-3, -1*E(48)^2-E(48)^-2, E(48)^10+E(48)^-10, E(48)^2+E(48)^-2, -1*E(48)^10-E(48)^-10, -1*E(48)^7-E(48)^-7, E(48)^11+E(48)^-11, -1*E(48)^5-E(48)^-5, -1*E(48)^11-E(48)^-11, E(48)^7+E(48)^-7, E(48)+E(48)^-1, -1*E(48)-E(48)^-1, E(48)^5+E(48)^-5], [2, -2, 0, 0, -1, 0, 1, E(48)^6+E(48)^-6, -1*E(48)^6-E(48)^-6, E(48)^4+E(48)^-4, -1*E(48)^4-E(48)^-4, -1*E(48)^3-E(48)^-3, -1*E(48)^9-E(48)^-9, E(48)^9+E(48)^-9, E(48)^3+E(48)^-3, E(48)^10+E(48)^-10, -1*E(48)^2-E(48)^-2, -1*E(48)^10-E(48)^-10, E(48)^2+E(48)^-2, -1*E(48)-E(48)^-1, -1*E(48)^5-E(48)^-5, E(48)^11+E(48)^-11, E(48)^5+E(48)^-5, E(48)+E(48)^-1, E(48)^7+E(48)^-7, -1*E(48)^7-E(48)^-7, -1*E(48)^11-E(48)^-11], [2, -2, 0, 0, -1, 0, 1, E(48)^6+E(48)^-6, -1*E(48)^6-E(48)^-6, E(48)^4+E(48)^-4, -1*E(48)^4-E(48)^-4, E(48)^3+E(48)^-3, E(48)^9+E(48)^-9, -1*E(48)^9-E(48)^-9, -1*E(48)^3-E(48)^-3, E(48)^10+E(48)^-10, -1*E(48)^2-E(48)^-2, -1*E(48)^10-E(48)^-10, E(48)^2+E(48)^-2, E(48)+E(48)^-1, E(48)^5+E(48)^-5, -1*E(48)^11-E(48)^-11, -1*E(48)^5-E(48)^-5, -1*E(48)-E(48)^-1, -1*E(48)^7-E(48)^-7, E(48)^7+E(48)^-7, E(48)^11+E(48)^-11]]; ConvertToLibraryCharacterTableNC(chartbl_96_6);