# Group 27.2 downloaded from the LMFDB on 01 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(34,27); a := GPC.1; b := GPC.2; GPerm := Group( (4,12,9,6,11,8,5,10,7), (1,3,2), (4,6,5)(7,9,8)(10,12,11) ); GLFp := Group([[[ Z(19)^4, 0*Z(19) ], [ 0*Z(19), Z(19)^16 ]], [[ Z(19)^6, 0*Z(19) ], [ 0*Z(19), Z(19)^12 ]]]); # Booleans booleans_27_2 := rec( Agroup := true, Zgroup := false, abelian := true, almost_simple := false, cyclic := false, metabelian := true, metacyclic := true, monomial := true, nilpotent := true, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_27_2:=rec(); chartbl_27_2.IsFinite:= true; chartbl_27_2.UnderlyingCharacteristic:= 0; chartbl_27_2.UnderlyingGroup:= GPC; chartbl_27_2.Size:= 27; chartbl_27_2.InfoText:= "Character table for group 27.2 downloaded from the LMFDB."; chartbl_27_2.Identifier:= " C3*C9 "; chartbl_27_2.NrConjugacyClasses:= 27; chartbl_27_2.ConjugacyClasses:= [ of ..., f1, f1^2, f1*f3, f1^2*f3^2, f1*f3^2, f1^2*f3, f3, f3^2, f2, f2^2*f3^2, f2^2, f2*f3^2, f2*f3, f2^2*f3, f1*f2, f1^2*f2^2*f3^2, f1^2*f2^2, f1*f2*f3^2, f1*f2*f3, f1^2*f2^2*f3, f1*f2^2, f1^2*f2*f3^2, f1^2*f2*f3, f1*f2^2*f3, f1*f2^2*f3^2, f1^2*f2]; chartbl_27_2.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_27_2.ComputedPowerMaps:= [ , [1, 3, 2, 5, 4, 7, 6, 9, 8, 12, 13, 14, 15, 11, 10, 18, 19, 20, 21, 17, 16, 24, 25, 26, 27, 23, 22]]; chartbl_27_2.SizesCentralizers:= [27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27]; chartbl_27_2.ClassNames:= ["1A", "3A1", "3A-1", "3B1", "3B-1", "3C1", "3C-1", "3D1", "3D-1", "9A1", "9A-1", "9A2", "9A-2", "9A4", "9A-4", "9B1", "9B-1", "9B2", "9B-2", "9B4", "9B-4", "9C1", "9C-1", "9C2", "9C-2", "9C4", "9C-4"]; chartbl_27_2.OrderClassRepresentatives:= [1, 3, 3, 3, 3, 3, 3, 3, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9]; chartbl_27_2.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, E(3)^-1, E(3)^-1, E(3), 1, E(3), E(3), E(3)^-1, E(3), E(3)^-1, E(3), E(3)^-1, E(3), 1, E(3), E(3)^-1, E(3)^-1, 1, E(3), 1, E(3)^-1, 1, E(3), 1, E(3)^-1, 1], [1, 1, E(3), E(3), E(3)^-1, 1, E(3)^-1, E(3)^-1, E(3), E(3)^-1, E(3), E(3)^-1, E(3), E(3)^-1, 1, E(3)^-1, E(3), E(3), 1, E(3)^-1, 1, E(3), 1, E(3)^-1, 1, E(3), 1], [1, 1, E(3)^-1, E(3)^-1, E(3), 1, E(3), E(3), E(3)^-1, E(3)^-1, 1, E(3)^-1, E(3), E(3)^-1, E(3)^-1, 1, E(3), 1, E(3)^-1, 1, E(3), E(3), E(3)^-1, 1, E(3), 1, E(3)], [1, 1, E(3), E(3), E(3)^-1, 1, E(3)^-1, E(3)^-1, E(3), E(3), 1, E(3), E(3)^-1, E(3), E(3), 1, E(3)^-1, 1, E(3), 1, E(3)^-1, E(3)^-1, E(3), 1, E(3)^-1, 1, E(3)^-1], [1, 1, E(3)^-1, E(3)^-1, E(3), 1, E(3), E(3), E(3)^-1, 1, E(3), 1, 1, 1, E(3), E(3)^-1, 1, E(3), E(3), E(3)^-1, E(3)^-1, 1, E(3), E(3)^-1, E(3)^-1, E(3), E(3)^-1], [1, 1, E(3), E(3), E(3)^-1, 1, E(3)^-1, E(3)^-1, E(3), 1, E(3)^-1, 1, 1, 1, E(3)^-1, E(3), 1, E(3)^-1, E(3)^-1, E(3), E(3), 1, E(3)^-1, E(3), E(3), E(3)^-1, E(3)], [1, 1, 1, 1, 1, 1, 1, 1, 1, E(3)^-1, E(3)^-1, E(3)^-1, E(3), E(3)^-1, E(3), E(3), E(3), E(3)^-1, E(3), E(3), E(3)^-1, E(3), E(3), E(3), E(3)^-1, E(3)^-1, E(3)^-1], [1, 1, 1, 1, 1, 1, 1, 1, 1, E(3), E(3), E(3), E(3)^-1, E(3), E(3)^-1, E(3)^-1, E(3)^-1, E(3), E(3)^-1, E(3)^-1, E(3), E(3)^-1, E(3)^-1, E(3)^-1, E(3), E(3), E(3)], [1, E(9)^-3, E(9)^3, E(9)^-3, E(9)^-3, E(9)^3, 1, E(9)^3, 1, E(9)^-4, E(9)^2, E(9)^-1, E(9)^4, E(9)^2, E(9)^4, E(9), E(9), E(9)^-4, E(9), E(9)^-2, E(9)^2, E(9)^-2, E(9)^-2, E(9)^4, E(9)^-1, E(9)^-1, E(9)^-4], [1, E(9)^3, E(9)^-3, E(9)^3, E(9)^3, E(9)^-3, 1, E(9)^-3, 1, E(9)^4, E(9)^-2, E(9), E(9)^-4, E(9)^-2, E(9)^-4, E(9)^-1, E(9)^-1, E(9)^4, E(9)^-1, E(9)^2, E(9)^-2, E(9)^2, E(9)^2, E(9)^-4, E(9), E(9), E(9)^4], [1, E(9)^-3, E(9)^3, E(9)^-3, E(9)^-3, E(9)^3, 1, E(9)^3, 1, E(9)^2, E(9)^-1, E(9)^-4, E(9)^-2, E(9)^-1, E(9)^-2, E(9)^4, E(9)^4, E(9)^2, E(9)^4, E(9), E(9)^-1, E(9), E(9), E(9)^-2, E(9)^-4, E(9)^-4, E(9)^2], [1, E(9)^3, E(9)^-3, E(9)^3, E(9)^3, E(9)^-3, 1, E(9)^-3, 1, E(9)^-2, E(9), E(9)^4, E(9)^2, E(9), E(9)^2, E(9)^-4, E(9)^-4, E(9)^-2, E(9)^-4, E(9)^-1, E(9), E(9)^-1, E(9)^-1, E(9)^2, E(9)^4, E(9)^4, E(9)^-2], [1, E(9)^-3, E(9)^3, E(9)^-3, E(9)^-3, E(9)^3, 1, E(9)^3, 1, E(9)^-1, E(9)^-4, E(9)^2, E(9), E(9)^-4, E(9), E(9)^-2, E(9)^-2, E(9)^-1, E(9)^-2, E(9)^4, E(9)^-4, E(9)^4, E(9)^4, E(9), E(9)^2, E(9)^2, E(9)^-1], [1, E(9)^3, E(9)^-3, E(9)^3, E(9)^3, E(9)^-3, 1, E(9)^-3, 1, E(9), E(9)^4, E(9)^-2, E(9)^-1, E(9)^4, E(9)^-1, E(9)^2, E(9)^2, E(9), E(9)^2, E(9)^-4, E(9)^4, E(9)^-4, E(9)^-4, E(9)^-1, E(9)^-2, E(9)^-2, E(9)], [1, E(9)^-3, E(9)^-3, 1, E(9)^3, E(9)^3, E(9)^-3, 1, E(9)^3, E(9)^-4, E(9)^-1, E(9)^-1, E(9)^4, E(9)^2, E(9), E(9)^4, E(9), E(9)^2, E(9)^-2, E(9), E(9)^-4, E(9)^-2, E(9)^4, E(9)^-2, E(9)^2, E(9)^-4, E(9)^-1], [1, E(9)^3, E(9)^3, 1, E(9)^-3, E(9)^-3, E(9)^3, 1, E(9)^-3, E(9)^4, E(9), E(9), E(9)^-4, E(9)^-2, E(9)^-1, E(9)^-4, E(9)^-1, E(9)^-2, E(9)^2, E(9)^-1, E(9)^4, E(9)^2, E(9)^-4, E(9)^2, E(9)^-2, E(9)^4, E(9)], [1, E(9)^-3, E(9)^-3, 1, E(9)^3, E(9)^3, E(9)^-3, 1, E(9)^3, E(9)^2, E(9)^-4, E(9)^-4, E(9)^-2, E(9)^-1, E(9)^4, E(9)^-2, E(9)^4, E(9)^-1, E(9), E(9)^4, E(9)^2, E(9), E(9)^-2, E(9), E(9)^-1, E(9)^2, E(9)^-4], [1, E(9)^3, E(9)^3, 1, E(9)^-3, E(9)^-3, E(9)^3, 1, E(9)^-3, E(9)^-2, E(9)^4, E(9)^4, E(9)^2, E(9), E(9)^-4, E(9)^2, E(9)^-4, E(9), E(9)^-1, E(9)^-4, E(9)^-2, E(9)^-1, E(9)^2, E(9)^-1, E(9), E(9)^-2, E(9)^4], [1, E(9)^-3, E(9)^-3, 1, E(9)^3, E(9)^3, E(9)^-3, 1, E(9)^3, E(9)^-1, E(9)^2, E(9)^2, E(9), E(9)^-4, E(9)^-2, E(9), E(9)^-2, E(9)^-4, E(9)^4, E(9)^-2, E(9)^-1, E(9)^4, E(9), E(9)^4, E(9)^-4, E(9)^-1, E(9)^2], [1, E(9)^3, E(9)^3, 1, E(9)^-3, E(9)^-3, E(9)^3, 1, E(9)^-3, E(9), E(9)^-2, E(9)^-2, E(9)^-1, E(9)^4, E(9)^2, E(9)^-1, E(9)^2, E(9)^4, E(9)^-4, E(9)^2, E(9), E(9)^-4, E(9)^-1, E(9)^-4, E(9)^4, E(9), E(9)^-2], [1, E(9)^-3, 1, E(9)^3, 1, E(9)^3, E(9)^3, E(9)^-3, E(9)^-3, E(9)^-4, E(9)^-4, E(9)^-1, E(9)^4, E(9)^2, E(9)^-2, E(9)^-2, E(9), E(9)^-1, E(9)^4, E(9)^4, E(9)^-1, E(9)^-2, E(9), E(9), E(9)^-4, E(9)^2, E(9)^2], [1, E(9)^3, 1, E(9)^-3, 1, E(9)^-3, E(9)^-3, E(9)^3, E(9)^3, E(9)^4, E(9)^4, E(9), E(9)^-4, E(9)^-2, E(9)^2, E(9)^2, E(9)^-1, E(9), E(9)^-4, E(9)^-4, E(9), E(9)^2, E(9)^-1, E(9)^-1, E(9)^4, E(9)^-2, E(9)^-2], [1, E(9)^-3, 1, E(9)^3, 1, E(9)^3, E(9)^3, E(9)^-3, E(9)^-3, E(9)^2, E(9)^2, E(9)^-4, E(9)^-2, E(9)^-1, E(9), E(9), E(9)^4, E(9)^-4, E(9)^-2, E(9)^-2, E(9)^-4, E(9), E(9)^4, E(9)^4, E(9)^2, E(9)^-1, E(9)^-1], [1, E(9)^3, 1, E(9)^-3, 1, E(9)^-3, E(9)^-3, E(9)^3, E(9)^3, E(9)^-2, E(9)^-2, E(9)^4, E(9)^2, E(9), E(9)^-1, E(9)^-1, E(9)^-4, E(9)^4, E(9)^2, E(9)^2, E(9)^4, E(9)^-1, E(9)^-4, E(9)^-4, E(9)^-2, E(9), E(9)], [1, E(9)^-3, 1, E(9)^3, 1, E(9)^3, E(9)^3, E(9)^-3, E(9)^-3, E(9)^-1, E(9)^-1, E(9)^2, E(9), E(9)^-4, E(9)^4, E(9)^4, E(9)^-2, E(9)^2, E(9), E(9), E(9)^2, E(9)^4, E(9)^-2, E(9)^-2, E(9)^-1, E(9)^-4, E(9)^-4], [1, E(9)^3, 1, E(9)^-3, 1, E(9)^-3, E(9)^-3, E(9)^3, E(9)^3, E(9), E(9), E(9)^-2, E(9)^-1, E(9)^4, E(9)^-4, E(9)^-4, E(9)^2, E(9)^-2, E(9)^-1, E(9)^-1, E(9)^-2, E(9)^-4, E(9)^2, E(9)^2, E(9), E(9)^4, E(9)^4]]; ConvertToLibraryCharacterTableNC(chartbl_27_2);