# Group 216.165 downloaded from the LMFDB on 31 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(8909543178979295278179679509,216); a := GPC.1; b := GPC.2; c := GPC.3; d := GPC.5;
GPerm := Group( (2,3)(6,7)(9,10), (2,3,4)(5,6,7)(8,9,10), (5,6,7), (5,6,7)(8,10,9), (1,2)(3,4), (1,3)(2,4) );
GLZN := Group([[[ZmodnZObj(1,36), ZmodnZObj(12,36)], [ZmodnZObj(0,36), ZmodnZObj(1,36)]],[[ZmodnZObj(19,36), ZmodnZObj(0,36)], [ZmodnZObj(18,36), ZmodnZObj(19,36)]],[[ZmodnZObj(31,36), ZmodnZObj(5,36)], [ZmodnZObj(6,36), ZmodnZObj(23,36)]],[[ZmodnZObj(1,36), ZmodnZObj(27,36)], [ZmodnZObj(27,36), ZmodnZObj(10,36)]],[[ZmodnZObj(13,36), ZmodnZObj(12,36)], [ZmodnZObj(24,36), ZmodnZObj(25,36)]],[[ZmodnZObj(1,36), ZmodnZObj(18,36)], [ZmodnZObj(18,36), ZmodnZObj(1,36)]]]);

 
# Booleans 
booleans_216_165 := 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 := true,
solvable := true,
supersolvable := false); 

 
# Character Table 
chartbl_216_165:=rec(); 
chartbl_216_165.IsFinite:= true; 
chartbl_216_165.UnderlyingCharacteristic:= 0; 
chartbl_216_165.UnderlyingGroup:= GPerm;
chartbl_216_165.Size:= 216;
chartbl_216_165.InfoText:= "Character table for group 216.165 downloaded from the LMFDB."; 
chartbl_216_165.Identifier:= " C3^2:S4 "; 
chartbl_216_165.NrConjugacyClasses:= 21; 
chartbl_216_165.ConjugacyClasses:= [(), (1,2)(3,4), (1,4)(5,7)(8,9), (8,10,9), (5,7,6), (5,7,6)(8,10,9), (5,7,6)(8,9,10), (2,3,4), (2,3,4)(8,9,10), (2,3,4)(8,10,9), (2,3,4)(5,6,7), (2,3,4)(5,6,7)(8,9,10), (2,3,4)(5,6,7)(8,10,9), (2,3,4)(5,7,6), (2,3,4)(5,7,6)(8,9,10), (2,3,4)(5,7,6)(8,10,9), (1,3,4,2)(6,7)(8,10), (1,2)(3,4)(8,9,10), (1,2)(3,4)(5,6,7), (1,2)(3,4)(5,6,7)(8,9,10), (1,2)(3,4)(5,6,7)(8,10,9)];
chartbl_216_165.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];
chartbl_216_165.ComputedPowerMaps:= [ , [1, 1, 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 2, 4, 5, 6, 7], [1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 2, 2, 2, 2]];
chartbl_216_165.SizesCentralizers:= [216, 72, 4, 108, 108, 108, 108, 27, 27, 27, 27, 27, 27, 27, 27, 27, 4, 36, 36, 36, 36];
chartbl_216_165.ClassNames:= ["1A", "2A", "2B", "3A", "3B", "3C", "3D", "3E", "3F", "3G", "3H", "3I", "3J", "3K", "3L", "3M", "4A", "6A", "6B", "6C", "6D"];
chartbl_216_165.OrderClassRepresentatives:= [1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 6, 6, 6, 6];
chartbl_216_165.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], [2, 2, 0, -1, -1, -1, 2, -1, -1, -1, 2, -1, -1, -1, 2, 2, 0, -1, -1, -1, 2], [2, 2, 0, -1, -1, -1, 2, -1, -1, 2, -1, 2, 2, -1, -1, -1, 0, -1, -1, -1, 2], [2, 2, 0, -1, -1, -1, 2, 2, 2, -1, -1, -1, -1, 2, -1, -1, 0, -1, -1, -1, 2], [2, 2, 0, -1, -1, 2, -1, -1, -1, 2, -1, -1, -1, 2, -1, 2, 0, -1, -1, 2, -1], [2, 2, 0, -1, -1, 2, -1, -1, 2, -1, -1, -1, 2, -1, 2, -1, 0, -1, -1, 2, -1], [2, 2, 0, -1, -1, 2, -1, 2, -1, -1, 2, 2, -1, -1, -1, -1, 0, -1, -1, 2, -1], [2, 2, 0, -1, 2, -1, -1, -1, -1, -1, -1, 2, -1, 2, 2, -1, 0, -1, 2, -1, -1], [2, 2, 0, -1, 2, -1, -1, -1, 2, 2, 2, -1, -1, -1, -1, -1, 0, -1, 2, -1, -1], [2, 2, 0, -1, 2, -1, -1, 2, -1, -1, -1, -1, 2, -1, -1, 2, 0, -1, 2, -1, -1], [2, 2, 0, 2, -1, -1, -1, -1, -1, -1, 2, -1, 2, 2, -1, -1, 0, 2, -1, -1, -1], [2, 2, 0, 2, -1, -1, -1, -1, 2, -1, -1, 2, -1, -1, -1, 2, 0, 2, -1, -1, -1], [2, 2, 0, 2, -1, -1, -1, 2, -1, 2, -1, -1, -1, -1, 2, -1, 0, 2, -1, -1, -1], [2, 2, 0, 2, 2, 2, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 2, 2, 2, 2], [3, -1, 1, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1], [3, -1, -1, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1], [6, -2, 0, -3, -3, -3, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, -2], [6, -2, 0, -3, -3, 6, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -2, 1], [6, -2, 0, -3, 6, -3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 1], [6, -2, 0, 6, -3, -3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 1, 1]];
ConvertToLibraryCharacterTableNC(chartbl_216_165);