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## G = C3⋊S32order 324 = 22·34

### Direct product of C3⋊S3 and C3⋊S3

Aliases: C3⋊S32, C3316D6, C345C22, C325S32, C34⋊C22C2, C32(S3×C3⋊S3), (C3×C3⋊S3)⋊4S3, C327(C2×C3⋊S3), (C32×C3⋊S3)⋊4C2, SmallGroup(324,169)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — C3⋊S32
 Chief series C1 — C3 — C32 — C33 — C34 — C32×C3⋊S3 — C3⋊S32
 Lower central C34 — C3⋊S32
 Upper central C1

Generators and relations for C3⋊S32
G = < a,b,c,d,e,f | a3=b3=c2=d3=e3=f2=1, ab=ba, cac=a-1, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 3412 in 356 conjugacy classes, 50 normal (4 characteristic)
C1, C2 [×3], C3 [×8], C3 [×16], C22, S3 [×32], C6 [×8], C32 [×18], C32 [×56], D6 [×8], C3×S3 [×32], C3⋊S3 [×2], C3⋊S3 [×74], C3×C6 [×2], C33 [×8], C33 [×16], S32 [×16], C2×C3⋊S3 [×2], S3×C32 [×8], C3×C3⋊S3 [×8], C33⋊C2 [×24], C34, S3×C3⋊S3 [×8], C32×C3⋊S3 [×2], C34⋊C2, C3⋊S32
Quotients: C1, C2 [×3], C22, S3 [×8], D6 [×8], C3⋊S3 [×2], S32 [×16], C2×C3⋊S3 [×2], S3×C3⋊S3 [×8], C3⋊S32

Permutation representations of C3⋊S32
On 18 points - transitive group 18T138
Generators in S18
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 8)(2 6 9)(3 4 7)(10 13 16)(11 14 17)(12 15 18)
(1 11)(2 10)(3 12)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)
(1 2 3)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 18 17)
(1 7 6)(2 8 4)(3 9 5)(10 14 18)(11 15 16)(12 13 17)
(1 11)(2 12)(3 10)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)```

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

`G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,8)(2,6,9)(3,4,7)(10,13,16)(11,14,17)(12,15,18), (1,11)(2,10)(3,12)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13), (1,2,3)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,18,17), (1,7,6)(2,8,4)(3,9,5)(10,14,18)(11,15,16)(12,13,17), (1,11)(2,12)(3,10)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18) );`

`G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,8),(2,6,9),(3,4,7),(10,13,16),(11,14,17),(12,15,18)], [(1,11),(2,10),(3,12),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13)], [(1,2,3),(4,5,6),(7,8,9),(10,12,11),(13,15,14),(16,18,17)], [(1,7,6),(2,8,4),(3,9,5),(10,14,18),(11,15,16),(12,13,17)], [(1,11),(2,12),(3,10),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18)])`

`G:=TransitiveGroup(18,138);`

36 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3X 6A ··· 6H order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 9 9 81 2 ··· 2 4 ··· 4 18 ··· 18

36 irreducible representations

 dim 1 1 1 2 2 4 type + + + + + + image C1 C2 C2 S3 D6 S32 kernel C3⋊S32 C32×C3⋊S3 C34⋊C2 C3×C3⋊S3 C33 C32 # reps 1 2 1 8 8 16

Matrix representation of C3⋊S32 in GL8(ℤ)

 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1

`G:=sub<GL(8,Integers())| [-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1] >;`

C3⋊S32 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3^2`
`% in TeX`

`G:=Group("C3:S3^2");`
`// GroupNames label`

`G:=SmallGroup(324,169);`
`// by ID`

`G=gap.SmallGroup(324,169);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,297,2164,7781]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^3=e^3=f^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;`
`// generators/relations`

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