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## G = C9⋊C36order 324 = 22·34

### The semidirect product of C9 and C36 acting via C36/C6=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C9⋊C36
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C2×C9⋊C9 — C9⋊C36
 Lower central C9 — C9⋊C36
 Upper central C1 — C6

Generators and relations for C9⋊C36
G = < a,b | a9=b36=1, bab-1=a5 >

Smallest permutation representation of C9⋊C36
On 36 points
Generators in S36
```(1 9 29 25 33 17 13 21 5)(2 18 10 14 30 22 26 6 34)(3 23 19 27 11 7 15 35 31)(4 8 24 16 20 36 28 32 12)
(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)| (1,9,29,25,33,17,13,21,5)(2,18,10,14,30,22,26,6,34)(3,23,19,27,11,7,15,35,31)(4,8,24,16,20,36,28,32,12), (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( (1,9,29,25,33,17,13,21,5)(2,18,10,14,30,22,26,6,34)(3,23,19,27,11,7,15,35,31)(4,8,24,16,20,36,28,32,12), (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([[(1,9,29,25,33,17,13,21,5),(2,18,10,14,30,22,26,6,34),(3,23,19,27,11,7,15,35,31),(4,8,24,16,20,36,28,32,12)], [(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 4A 4B 6A 6B 6C 6D 6E 9A ··· 9F 9G ··· 9O 12A 12B 12C 12D 18A ··· 18F 18G ··· 18O 36A ··· 36L order 1 2 3 3 3 3 3 4 4 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 9 9 1 1 2 2 2 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 C9⋊C6 C9⋊C12 C9⋊C18 C9⋊C36 kernel C9⋊C36 C2×C9⋊C9 C3×Dic9 C9⋊C9 C3×C18 Dic9 C3×C9 C18 C9 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 C9⋊C36 in GL6(𝔽37)

 10 11 27 0 0 0 0 0 26 0 0 0 9 11 27 0 0 0 0 0 0 1 0 36 0 0 0 16 36 11 0 0 0 0 10 0
,
 0 0 0 10 1 26 0 0 0 12 27 36 0 0 0 0 1 0 27 36 11 0 0 0 25 10 1 0 0 0 0 36 0 0 0 0

`G:=sub<GL(6,GF(37))| [10,0,9,0,0,0,11,0,11,0,0,0,27,26,27,0,0,0,0,0,0,1,16,0,0,0,0,0,36,10,0,0,0,36,11,0],[0,0,0,27,25,0,0,0,0,36,10,36,0,0,0,11,1,0,10,12,0,0,0,0,1,27,1,0,0,0,26,36,0,0,0,0] >;`

C9⋊C36 in GAP, Magma, Sage, TeX

`C_9\rtimes C_{36}`
`% in TeX`

`G:=Group("C9:C36");`
`// GroupNames label`

`G:=SmallGroup(324,9);`
`// by ID`

`G=gap.SmallGroup(324,9);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,5404,2170,208,7781]);`
`// Polycyclic`

`G:=Group<a,b|a^9=b^36=1,b*a*b^-1=a^5>;`
`// generators/relations`

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