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## G = C32⋊C36order 324 = 22·34

### The semidirect product of C32 and C36 acting via C36/C6=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C36
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C2×C32⋊C9 — C32⋊C36
 Lower central C32 — C32⋊C36
 Upper central C1 — C6

Generators and relations for C32⋊C36
G = < a,b,c | a3=b3=c36=1, ab=ba, cac-1=a-1b, cbc-1=b-1 >

Smallest permutation representation of C32⋊C36
On 36 points
Generators in S36
```(2 26 14)(3 27 15)(5 17 29)(6 18 30)(8 32 20)(9 33 21)(11 23 35)(12 24 36)
(1 25 13)(2 14 26)(3 27 15)(4 16 28)(5 29 17)(6 18 30)(7 31 19)(8 20 32)(9 33 21)(10 22 34)(11 35 23)(12 24 36)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)```

`G:=sub<Sym(36)| (2,26,14)(3,27,15)(5,17,29)(6,18,30)(8,32,20)(9,33,21)(11,23,35)(12,24,36), (1,25,13)(2,14,26)(3,27,15)(4,16,28)(5,29,17)(6,18,30)(7,31,19)(8,20,32)(9,33,21)(10,22,34)(11,35,23)(12,24,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)>;`

`G:=Group( (2,26,14)(3,27,15)(5,17,29)(6,18,30)(8,32,20)(9,33,21)(11,23,35)(12,24,36), (1,25,13)(2,14,26)(3,27,15)(4,16,28)(5,29,17)(6,18,30)(7,31,19)(8,20,32)(9,33,21)(10,22,34)(11,35,23)(12,24,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36) );`

`G=PermutationGroup([[(2,26,14),(3,27,15),(5,17,29),(6,18,30),(8,32,20),(9,33,21),(11,23,35),(12,24,36)], [(1,25,13),(2,14,26),(3,27,15),(4,16,28),(5,29,17),(6,18,30),(7,31,19),(8,20,32),(9,33,21),(10,22,34),(11,35,23),(12,24,36)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)]])`

60 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 18A ··· 18F 18G ··· 18L 36A ··· 36L order 1 2 3 3 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 6 6 6 9 9 1 1 2 2 2 6 6 6 3 ··· 3 6 ··· 6 9 9 9 9 3 ··· 3 6 ··· 6 9 ··· 9

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 6 6 6 6 type + + + - + - image C1 C2 C3 C4 C6 C9 C12 C18 C36 S3 Dic3 C3×S3 C3×Dic3 S3×C9 C9×Dic3 C32⋊C6 C32⋊C12 C32⋊C18 C32⋊C36 kernel C32⋊C36 C2×C32⋊C9 C3×C3⋊Dic3 C32⋊C9 C32×C6 C3⋊Dic3 C33 C3×C6 C32 C3×C18 C3×C9 C3×C6 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 1 1 2 2 6 6 1 1 2 2

Matrix representation of C32⋊C36 in GL6(𝔽37)

 1 0 0 0 0 0 0 26 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 26
,
 26 0 0 0 0 0 0 26 0 0 0 0 0 0 26 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 0 0 0 0 26 0 0 0 0 0 0 26 0 0 0 1 0 0 0 11 0 0 0 0 0 0 11 0 0 0 36 0 0 0 0 0

`G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,26,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,26],[26,0,0,0,0,0,0,26,0,0,0,0,0,0,26,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[0,0,0,0,0,36,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0,0,26,0,0,0,0,0,0,26,0,0,0,0] >;`

C32⋊C36 in GAP, Magma, Sage, TeX

`C_3^2\rtimes C_{36}`
`% in TeX`

`G:=Group("C3^2:C36");`
`// GroupNames label`

`G:=SmallGroup(324,7);`
`// by ID`

`G=gap.SmallGroup(324,7);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,2164,2170,7781]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^3=c^36=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=b^-1>;`
`// generators/relations`

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