# Group 324.41 downloaded from the LMFDB on 19 July 2026. 
 
## 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(15900180483150431225996912794255,324); a := GPC.1; b := GPC.2; c := GPC.4; d := GPC.6;
GPerm := Group( (4,8)(5,9)(6,7)(13,17)(14,18)(15,16)(22,26)(23,27)(24,25), (2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22), (1,21,12,3,20,11,2,19,10)(4,24,15,6,23,14,5,22,13)(7,27,18,9,26,17,8,25,16), (4,5,6)(7,9,8)(10,16,15)(11,17,13)(12,18,14)(19,22,26)(20,23,27)(21,24,25), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26) );

 
# Booleans 
booleans_324_41 := rec( 
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := false,
metacyclic := false,
monomial := true,
nilpotent := false,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := true); 

 
# Character Table 
chartbl_324_41:=rec(); 
chartbl_324_41.IsFinite:= true; 
chartbl_324_41.UnderlyingCharacteristic:= 0; 
chartbl_324_41.UnderlyingGroup:= GPC;
chartbl_324_41.Size:= 324;
chartbl_324_41.InfoText:= "Character table for group 324.41 downloaded from the LMFDB."; 
chartbl_324_41.Identifier:= " C3^2.S3^2 "; 
chartbl_324_41.NrConjugacyClasses:= 17; 
chartbl_324_41.ConjugacyClasses:= [<identity> of ..., f1, f1*f2*f4*f6, f2*f3*f6, f5^2, f6^2, f3^2*f5, f3^2*f4*f6, f1*f5, f1*f2*f4*f6^2, f2*f3^2*f5*f6, f4^2, f4*f5, f4^2*f5^2, f1*f4, f1*f4*f5, f1*f4^2];
chartbl_324_41.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];
chartbl_324_41.ComputedPowerMaps:= [ , [1, 1, 1, 1, 5, 6, 7, 8, 5, 6, 7, 13, 14, 12, 12, 14, 13], [1, 2, 3, 4, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 9, 9, 9]];
chartbl_324_41.SizesCentralizers:= [324, 36, 12, 12, 162, 54, 18, 9, 18, 6, 6, 54, 54, 54, 18, 18, 18];
chartbl_324_41.ClassNames:= ["1A", "2A", "2B", "2C", "3A", "3B", "3C", "3D", "6A", "6B", "6C", "9A1", "9A2", "9A4", "18A1", "18A5", "18A7"];
chartbl_324_41.OrderClassRepresentatives:= [1, 2, 2, 2, 3, 3, 3, 3, 6, 6, 6, 9, 9, 9, 18, 18, 18];
chartbl_324_41.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], [2, 0, 0, 2, 2, 2, -1, -1, 0, 0, -1, 2, 2, 2, 0, 0, 0], [2, 2, 0, 0, 2, 2, 2, -1, 2, 0, 0, -1, -1, -1, -1, -1, -1], [2, -2, 0, 0, 2, 2, 2, -1, -2, 0, 0, -1, -1, -1, 1, 1, 1], [2, 0, 0, -2, 2, 2, -1, -1, 0, 0, 1, 2, 2, 2, 0, 0, 0], [4, 0, 0, 0, 4, 4, -2, 1, 0, 0, 0, -2, -2, -2, 0, 0, 0], [6, 0, 2, 0, 6, -3, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [6, 0, -2, 0, 6, -3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [6, 2, 0, 0, -3, 0, 0, 0, -1, 0, 0, E(9)-E(9)^2+2*E(9)^4+E(9)^-4, E(9)-E(9)^2-E(9)^4-2*E(9)^-4, -2*E(9)+2*E(9)^2-E(9)^4+E(9)^-4, E(9)^4+E(9)^-4, E(9)^2+E(9)^-2, E(9)+E(9)^-1], [6, 2, 0, 0, -3, 0, 0, 0, -1, 0, 0, -2*E(9)+2*E(9)^2-E(9)^4+E(9)^-4, E(9)-E(9)^2+2*E(9)^4+E(9)^-4, E(9)-E(9)^2-E(9)^4-2*E(9)^-4, E(9)^2+E(9)^-2, E(9)+E(9)^-1, E(9)^4+E(9)^-4], [6, 2, 0, 0, -3, 0, 0, 0, -1, 0, 0, E(9)-E(9)^2-E(9)^4-2*E(9)^-4, -2*E(9)+2*E(9)^2-E(9)^4+E(9)^-4, E(9)-E(9)^2+2*E(9)^4+E(9)^-4, E(9)+E(9)^-1, E(9)^4+E(9)^-4, E(9)^2+E(9)^-2], [6, -2, 0, 0, -3, 0, 0, 0, 1, 0, 0, E(9)-E(9)^2+2*E(9)^4+E(9)^-4, E(9)-E(9)^2-E(9)^4-2*E(9)^-4, -2*E(9)+2*E(9)^2-E(9)^4+E(9)^-4, -1*E(9)^4-E(9)^-4, -1*E(9)^2-E(9)^-2, -1*E(9)-E(9)^-1], [6, -2, 0, 0, -3, 0, 0, 0, 1, 0, 0, -2*E(9)+2*E(9)^2-E(9)^4+E(9)^-4, E(9)-E(9)^2+2*E(9)^4+E(9)^-4, E(9)-E(9)^2-E(9)^4-2*E(9)^-4, -1*E(9)^2-E(9)^-2, -1*E(9)-E(9)^-1, -1*E(9)^4-E(9)^-4], [6, -2, 0, 0, -3, 0, 0, 0, 1, 0, 0, E(9)-E(9)^2-E(9)^4-2*E(9)^-4, -2*E(9)+2*E(9)^2-E(9)^4+E(9)^-4, E(9)-E(9)^2+2*E(9)^4+E(9)^-4, -1*E(9)-E(9)^-1, -1*E(9)^4-E(9)^-4, -1*E(9)^2-E(9)^-2]];
ConvertToLibraryCharacterTableNC(chartbl_324_41);