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## G = C92⋊8C6order 486 = 2·35

### 8th semidirect product of C92 and C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C92⋊8C6
 Chief series C1 — C3 — C9 — C3×C9 — C92 — C92⋊9C3 — C92⋊8C6
 Lower central C9 — C3×C9 — C92⋊8C6
 Upper central C1 — C3 — C9

Generators and relations for C928C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a7, cbc-1=b2 >

Subgroups: 256 in 71 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C3×D9, S3×C9, C9⋊C6, C2×3- 1+2, S3×C32, C92, C9⋊C9, C9⋊C9, C3×3- 1+2, C3×3- 1+2, C9×D9, C9⋊C18, C3×C9⋊C6, S3×3- 1+2, C929C3, C928C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, 3- 1+2, C9⋊C6, C2×3- 1+2, S3×C32, C3×C9⋊C6, S3×3- 1+2, C928C6

Permutation representations of C928C6
On 18 points - transitive group 18T158
Generators in S18
```(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 9 8 7 6 5 4 3 2)(10 11 12 13 14 15 16 17 18)
(1 16)(2 11 8 17 5 14)(3 15 6 18 9 12)(4 10)(7 13)```

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

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

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

`G:=TransitiveGroup(18,158);`

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 6A 6B 6C 6D 9A 9B 9C ··· 9M 9N 9O 9P 9Q 9R ··· 9W 18A ··· 18F order 1 2 3 3 3 3 3 3 3 6 6 6 6 9 9 9 ··· 9 9 9 9 9 9 ··· 9 18 ··· 18 size 1 9 1 1 2 2 2 9 9 9 9 27 27 3 3 6 ··· 6 9 9 9 9 18 ··· 18 27 ··· 27

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 6 6 6 6 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 S3 C3×S3 C3×S3 3- 1+2 C2×3- 1+2 C9⋊C6 C3×C9⋊C6 S3×3- 1+2 C92⋊8C6 kernel C92⋊8C6 C92⋊9C3 C9×D9 C9⋊C18 C3×C9⋊C6 C92 C9⋊C9 C3×3- 1+2 C3×3- 1+2 C3×C9 C33 D9 C9 C9 C3 C3 C1 # reps 1 1 2 4 2 2 4 2 1 6 2 2 2 1 2 2 6

Matrix representation of C928C6 in GL6(𝔽19)

 1 0 6 0 0 0 8 0 18 0 0 0 8 11 18 0 0 0 0 0 1 0 0 1 11 0 1 11 0 0 11 0 1 0 11 0
,
 7 4 0 0 0 0 0 12 7 0 0 0 18 12 0 0 0 0 1 7 0 0 0 1 12 7 0 11 0 0 12 7 0 0 11 0
,
 18 0 0 6 0 0 18 0 0 18 7 0 11 0 0 18 0 11 0 0 0 1 0 0 1 7 0 1 0 0 8 0 11 1 0 0

`G:=sub<GL(6,GF(19))| [1,8,8,0,11,11,0,0,11,0,0,0,6,18,18,1,1,1,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[7,0,18,1,12,12,4,12,12,7,7,7,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[18,18,11,0,1,8,0,0,0,0,7,0,0,0,0,0,0,11,6,18,18,1,1,1,0,7,0,0,0,0,0,0,11,0,0,0] >;`

C928C6 in GAP, Magma, Sage, TeX

`C_9^2\rtimes_8C_6`
`% in TeX`

`G:=Group("C9^2:8C6");`
`// GroupNames label`

`G:=SmallGroup(486,110);`
`// by ID`

`G=gap.SmallGroup(486,110);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,122,8104,3250,208,11669]);`
`// Polycyclic`

`G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^2>;`
`// generators/relations`

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