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G = C32⋊C9⋊S3  order 486 = 2·35

1st semidirect product of C32⋊C9 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C32⋊C9⋊1S3, (C3×He3).1C6, C33.4(C3×S3), He3⋊5S3.1C3, C3.4(C33⋊C6), C33.C32⋊3C2, C32.28(C32⋊C6), C3.1(He3.C6), SmallGroup(486,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C3×He3 — C32⋊C9⋊S3
Chief series
C1 — C3 — C32 — C33 — C3×He3 — C33.C32 — C32⋊C9⋊S3
Lower central
C3×He3 — C32⋊C9⋊S3
Upper central
C1 — C3

Generators and relations for C32⋊C9⋊S3
 G = < a,b,c,d,e | a3=b3=c9=d3=e2=1, ab=ba, cac-1=ab-1, dad-1=ac6, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=a-1bc7, ce=ec, ede=d-1 >

Subgroups: 542 in 59 conjugacy classes, 10 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, He3, 3- 1+2, C33, C33, S3×C9, He3⋊C2, C3×C3⋊S3, C32⋊C9, C3×He3, C3×3- 1+2, C32⋊C18, He3⋊5S3, C33.C32, C32⋊C9⋊S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C32⋊C6, C33⋊C6, He3.C6, C32⋊C9⋊S3

Permutation representations of C32⋊C9⋊S3
►On 18 points - transitive group 18T173
Generators in S18
(1 7 4)(2 5 8)(10 13 16)(11 17 14)

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

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

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

(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)
 
G:=sub<Sym(18)| (1,7,4)(2,5,8)(10,13,16)(11,17,14), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,2,9)(3,4,5)(6,7,8)(10,18,11)(12,14,13)(15,17,16), (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15)>;
 
G:=Group( (1,7,4)(2,5,8)(10,13,16)(11,17,14), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,2,9)(3,4,5)(6,7,8)(10,18,11)(12,14,13)(15,17,16), (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15) );
 
G=PermutationGroup([[(1,7,4),(2,5,8),(10,13,16),(11,17,14)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,2,9),(3,4,5),(6,7,8),(10,18,11),(12,14,13),(15,17,16)], [(1,16),(2,17),(3,18),(4,10),(5,11),(6,12),(7,13),(8,14),(9,15)]])
 
G:=TransitiveGroup(18,173);
 

31 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H3I6A6B9A···9F9G···9L18A···18F
order12333333333669···99···918···18
size127112221818181827279···918···1827···27

31 irreducible representations

dim1111223666
type+++++
imageC1C2C3C6S3C3×S3He3.C6C32⋊C6C33⋊C6C32⋊C9⋊S3
kernelC32⋊C9⋊S3C33.C32He3⋊5S3C3×He3C32⋊C9C33C3C32C3C1
# reps11221212136

Matrix representation of C32⋊C9⋊S3 ►in GL6(𝔽19)

700000
8110000
701000
0001100
0001170
0001801
,
1100000
0110000
0011000
000700
000070
000007
,
1010000
0018000
11118000
0001010
0000018
00011118
,
1010000
0018000
0118000
0001100
0000181
0000180
,
000100
000010
000001
100000
010000
001000

G:=sub<GL(6,GF(19))| [7,8,7,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,11,11,18,0,0,0,0,7,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,1,0,0,0,0,0,11,0,0,0,10,18,18,0,0,0,0,0,0,1,0,1,0,0,0,0,0,11,0,0,0,10,18,18],[1,0,0,0,0,0,0,0,1,0,0,0,10,18,18,0,0,0,0,0,0,1,0,0,0,0,0,10,18,18,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
 

C32⋊C9⋊S3 in GAP, Magma, Sage, TeX 

C_3^2\rtimes C_9\rtimes S_3
 
% in TeX
 
G:=Group("C3^2:C9:S3");
 
// GroupNames label
 
G:=SmallGroup(486,7);
 
// by ID
 
G=gap.SmallGroup(486,7);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,218,224,8643,873,1383,3244]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^9=d^3=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*c^6,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=a^-1*b*c^7,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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