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G = C32⋊C9⋊C6  order 486 = 2·35

1st semidirect product of C32⋊C9 and C6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C3.2C3≀S3, C32⋊C9⋊1C6, (C3×He3).1S3, C33.3(C3×S3), C32⋊2D9⋊2C3, C33.C32⋊2C2, C32.27(C32⋊C6), C3.3(He3.S3), SmallGroup(486,6)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C32⋊C9 — C32⋊C9⋊C6
Chief series
C1 — C3 — C32 — C33 — C32⋊C9 — C33.C32 — C32⋊C9⋊C6
Lower central
C32⋊C9 — C32⋊C9⋊C6
Upper central
C1 — C3

Generators and relations for C32⋊C9⋊C6
 G = < a,b,c,d | a3=b3=c9=d6=1, ab=ba, cac-1=ab-1, dad-1=a-1bc6, bc=cb, bd=db, dcd-1=a-1c5 >

Subgroups: 470 in 59 conjugacy classes, 10 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, He3, 3- 1+2, C33, C33, C3×D9, C32⋊C6, S3×C32, C3×C3⋊S3, C32⋊C9, C3×He3, C3×3- 1+2, C32⋊2D9, C3×C32⋊C6, C33.C32, C32⋊C9⋊C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C32⋊C6, C3≀S3, He3.S3, C32⋊C9⋊C6

Permutation representations of C32⋊C9⋊C6
►On 18 points - transitive group 18T174
Generators in S18
(1 7 4)(3 6 9)(11 17 14)(12 15 18)

(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)

(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

(1 15 9 16 2 17)(3 13 5 14 4 12)(6 10 8 11 7 18)
 
G:=sub<Sym(18)| (1,7,4)(3,6,9)(11,17,14)(12,15,18), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,15,9,16,2,17)(3,13,5,14,4,12)(6,10,8,11,7,18)>;
 
G:=Group( (1,7,4)(3,6,9)(11,17,14)(12,15,18), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,15,9,16,2,17)(3,13,5,14,4,12)(6,10,8,11,7,18) );
 
G=PermutationGroup([[(1,7,4),(3,6,9),(11,17,14),(12,15,18)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,15,9,16,2,17),(3,13,5,14,4,12),(6,10,8,11,7,18)]])
 
G:=TransitiveGroup(18,174);
 

31 conjugacy classes

class 1  2 3A3B3C3D3E3F···3K3L6A···6H9A···9I
order12333333···336···69···9
size127112229···91827···2718···18

31 irreducible representations

dim1111223666
type+++++
imageC1C2C3C6S3C3×S3C3≀S3C32⋊C6He3.S3C32⋊C9⋊C6
kernelC32⋊C9⋊C6C33.C32C32⋊2D9C32⋊C9C3×He3C33C3C32C3C1
# reps11221212136

Matrix representation of C32⋊C9⋊C6 ►in GL6(𝔽19)

700000
0110000
001000
000100
0000110
000007
,
700000
070000
007000
000700
000070
000007
,
070000
007000
100000
000001
0001100
0000110
,
000070
0000011
000100
070000
0011000
100000

G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,1,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0,7,0,0,0,0,0,0,11,0,0,0,1,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0] >;
 

C32⋊C9⋊C6 in GAP, Magma, Sage, TeX 

C_3^2\rtimes C_9\rtimes C_6
 
% in TeX
 
G:=Group("C3^2:C9:C6");
 
// GroupNames label
 
G:=SmallGroup(486,6);
 
// by ID
 
G=gap.SmallGroup(486,6);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,2162,224,176,4755,873,735,3244]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^9=d^6=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b*c^6,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^5>;
 
// generators/relations
 

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