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## G = C33⋊2A4order 324 = 22·34

### 1st semidirect product of C33 and A4 acting via A4/C22=C3

Aliases: C332A4, C62.18C32, C223C3≀C3, C32⋊A44C3, (C3×C62)⋊1C3, C32.A46C3, (C2×C6).12He3, C32.13(C3×A4), C3.12(C32⋊A4), SmallGroup(324,60)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C33⋊2A4
 Chief series C1 — C22 — C2×C6 — C62 — C32⋊A4 — C33⋊2A4
 Lower central C22 — C2×C6 — C62 — C33⋊2A4
 Upper central C1 — C3 — C32 — C33

Generators and relations for C332A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, faf-1=ab-1c, bc=cb, bd=db, be=eb, fbf-1=bc-1, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 250 in 68 conjugacy classes, 12 normal (10 characteristic)
C1, C2, C3, C3 [×5], C22, C6 [×13], C9 [×2], C32, C32 [×5], A4, C2×C6, C2×C6 [×4], C3×C6 [×13], He3, 3- 1+2 [×2], C33, C3.A4 [×2], C3×A4, C62, C62 [×4], C32×C6, C3≀C3, C32.A4 [×2], C32⋊A4, C3×C62, C332A4
Quotients: C1, C3 [×4], C32, A4, He3, C3×A4, C3≀C3, C32⋊A4, C332A4

Permutation representations of C332A4
On 18 points - transitive group 18T127
Generators in S18
```(13 14 15)(16 17 18)
(7 11 9)(8 12 10)(13 15 14)(16 18 17)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 14 15)(16 17 18)
(1 2)(3 4)(5 6)(13 16)(14 17)(15 18)
(7 8)(9 10)(11 12)(13 16)(14 17)(15 18)
(1 13 7)(2 16 8)(3 15 9)(4 18 10)(5 14 11)(6 17 12)```

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

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

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

`G:=TransitiveGroup(18,127);`

44 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 3K 3L 6A ··· 6Z 9A 9B 9C 9D order 1 2 3 3 3 ··· 3 3 3 6 ··· 6 9 9 9 9 size 1 3 1 1 3 ··· 3 36 36 3 ··· 3 36 36 36 36

44 irreducible representations

 dim 1 1 1 1 3 3 3 3 3 3 type + + image C1 C3 C3 C3 A4 He3 C3×A4 C3≀C3 C32⋊A4 C33⋊2A4 kernel C33⋊2A4 C32.A4 C32⋊A4 C3×C62 C33 C2×C6 C32 C22 C3 C1 # reps 1 4 2 2 1 2 2 6 6 18

Matrix representation of C332A4 in GL3(𝔽7) generated by

 1 0 0 0 2 1 0 2 3
,
 1 0 0 5 1 4 2 1 5
,
 4 0 0 0 4 0 0 0 4
,
 6 0 0 0 2 3 0 6 5
,
 6 0 0 1 5 4 6 1 2
,
 3 4 5 3 0 3 0 0 4
`G:=sub<GL(3,GF(7))| [1,0,0,0,2,2,0,1,3],[1,5,2,0,1,1,0,4,5],[4,0,0,0,4,0,0,0,4],[6,0,0,0,2,6,0,3,5],[6,1,6,0,5,1,0,4,2],[3,3,0,4,0,0,5,3,4] >;`

C332A4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_2A_4`
`% in TeX`

`G:=Group("C3^3:2A4");`
`// GroupNames label`

`G:=SmallGroup(324,60);`
`// by ID`

`G=gap.SmallGroup(324,60);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,650,4864,8753]);`
`// Polycyclic`

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

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