# Group 162.13 downloaded from the LMFDB on 17 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(180175900447401868591,162); a := GPC.1; b := GPC.3; c := GPC.4;
GPerm := Group( (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), (4,5,6)(7,9,8)(10,13,17)(11,14,18)(12,15,16)(19,25,24)(20,26,22)(21,27,23), (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), (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_162_13 := rec( 
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := true,
metacyclic := false,
monomial := true,
nilpotent := false,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := true); 

 
# Character Table 
chartbl_162_13:=rec(); 
chartbl_162_13.IsFinite:= true; 
chartbl_162_13.UnderlyingCharacteristic:= 0; 
chartbl_162_13.UnderlyingGroup:= GPC;
chartbl_162_13.Size:= 162;
chartbl_162_13.InfoText:= "Character table for group 162.13 downloaded from the LMFDB."; 
chartbl_162_13.Identifier:= " He3.S3 "; 
chartbl_162_13.NrConjugacyClasses:= 13; 
chartbl_162_13.ConjugacyClasses:= [<identity> of ..., f1*f2*f3^2*f4^2, f5, f3, f2^2*f3^2*f5^2, f2*f3, f1*f2^2*f3*f4^2, f1*f4^2*f5, f4, f4^2, f4*f5, f2^2*f3*f4, f2*f4^2*f5^2];
chartbl_162_13.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];
chartbl_162_13.ComputedPowerMaps:= [ , [1, 1, 3, 4, 6, 5, 5, 6, 10, 11, 9, 13, 12], [1, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3]];
chartbl_162_13.SizesCentralizers:= [162, 6, 81, 27, 18, 18, 6, 6, 27, 27, 27, 9, 9];
chartbl_162_13.ClassNames:= ["1A", "2A", "3A", "3B", "3C1", "3C-1", "6A1", "6A-1", "9A1", "9A2", "9A4", "9B1", "9B-1"];
chartbl_162_13.OrderClassRepresentatives:= [1, 2, 3, 3, 3, 3, 6, 6, 9, 9, 9, 9, 9];
chartbl_162_13.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, E(3)^-1, E(3), E(3), E(3)^-1, 1, 1, 1, E(3)^-1, E(3)], [1, 1, 1, 1, E(3), E(3)^-1, E(3)^-1, E(3), 1, 1, 1, E(3), E(3)^-1], [1, -1, 1, 1, E(3)^-1, E(3), -1*E(3), -1*E(3)^-1, 1, 1, 1, E(3)^-1, E(3)], [1, -1, 1, 1, E(3), E(3)^-1, -1*E(3)^-1, -1*E(3), 1, 1, 1, E(3), E(3)^-1], [2, 0, 2, 2, 2, 2, 0, 0, -1, -1, -1, -1, -1], [2, 0, 2, 2, 2*E(3)^-1, 2*E(3), 0, 0, -1, -1, -1, -1*E(3)^-1, -1*E(3)], [2, 0, 2, 2, 2*E(3), 2*E(3)^-1, 0, 0, -1, -1, -1, -1*E(3), -1*E(3)^-1], [6, 0, 6, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, -3, 0, 0, 0, 0, 0, -1*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)+E(9)^2-2*E(9)^4-E(9)^-4, 0, 0], [6, 0, -3, 0, 0, 0, 0, 0, 2*E(9)-2*E(9)^2+E(9)^4-E(9)^-4, -1*E(9)+E(9)^2-2*E(9)^4-E(9)^-4, -1*E(9)+E(9)^2+E(9)^4+2*E(9)^-4, 0, 0], [6, 0, -3, 0, 0, 0, 0, 0, -1*E(9)+E(9)^2-2*E(9)^4-E(9)^-4, -1*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, 0, 0]];
ConvertToLibraryCharacterTableNC(chartbl_162_13);