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## G = C9⋊C54order 486 = 2·35

### The semidirect product of C9 and C54 acting via C54/C9=C6

Aliases: C9⋊C54, D9⋊C27, C92.C6, C9⋊C27⋊C2, (C3×D9).C9, (C9×D9).C3, C9.5(C9⋊C6), C3.3(S3×C27), (C3×C9).1C18, (C3×C27).1S3, C3.3(C9⋊C18), C32.14(S3×C9), (C3×C9).49(C3×S3), SmallGroup(486,30)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C9⋊C54
 Chief series C1 — C3 — C9 — C3×C9 — C92 — C9⋊C27 — C9⋊C54
 Lower central C9 — C9⋊C54
 Upper central C1 — C9

Generators and relations for C9⋊C54
G = < a,b | a9=b54=1, bab-1=a2 >

Smallest permutation representation of C9⋊C54
On 54 points
Generators in S54
```(1 49 43 37 31 25 19 13 7)(2 44 32 20 8 50 38 26 14)(3 33 9 39 15 45 21 51 27)(4 10 16 22 28 34 40 46 52)(5 17 29 41 53 11 23 35 47)(6 30 54 24 48 18 42 12 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)```

`G:=sub<Sym(54)| (1,49,43,37,31,25,19,13,7)(2,44,32,20,8,50,38,26,14)(3,33,9,39,15,45,21,51,27)(4,10,16,22,28,34,40,46,52)(5,17,29,41,53,11,23,35,47)(6,30,54,24,48,18,42,12,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)>;`

`G:=Group( (1,49,43,37,31,25,19,13,7)(2,44,32,20,8,50,38,26,14)(3,33,9,39,15,45,21,51,27)(4,10,16,22,28,34,40,46,52)(5,17,29,41,53,11,23,35,47)(6,30,54,24,48,18,42,12,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54) );`

`G=PermutationGroup([[(1,49,43,37,31,25,19,13,7),(2,44,32,20,8,50,38,26,14),(3,33,9,39,15,45,21,51,27),(4,10,16,22,28,34,40,46,52),(5,17,29,41,53,11,23,35,47),(6,30,54,24,48,18,42,12,36)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)]])`

90 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 9A ··· 9F 9G ··· 9L 9M ··· 9U 18A ··· 18F 27A ··· 27R 27S ··· 27AJ 54A ··· 54R order 1 2 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 9 1 1 2 2 2 9 9 1 ··· 1 2 ··· 2 6 ··· 6 9 ··· 9 3 ··· 3 6 ··· 6 9 ··· 9

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 6 6 6 type + + + + image C1 C2 C3 C6 C9 C18 C27 C54 S3 C3×S3 S3×C9 S3×C27 C9⋊C6 C9⋊C18 C9⋊C54 kernel C9⋊C54 C9⋊C27 C9×D9 C92 C3×D9 C3×C9 D9 C9 C3×C27 C3×C9 C32 C3 C9 C3 C1 # reps 1 1 2 2 6 6 18 18 1 2 6 18 1 2 6

Matrix representation of C9⋊C54 in GL6(𝔽109)

 75 0 0 0 0 0 0 38 0 0 0 0 0 0 105 0 0 0 0 0 0 16 0 0 0 0 0 0 66 0 0 0 0 0 0 27
,
 0 0 0 0 0 27 0 0 0 105 0 0 0 0 0 0 105 0 0 0 27 0 0 0 105 0 0 0 0 0 0 105 0 0 0 0

`G:=sub<GL(6,GF(109))| [75,0,0,0,0,0,0,38,0,0,0,0,0,0,105,0,0,0,0,0,0,16,0,0,0,0,0,0,66,0,0,0,0,0,0,27],[0,0,0,0,105,0,0,0,0,0,0,105,0,0,0,27,0,0,0,105,0,0,0,0,0,0,105,0,0,0,27,0,0,0,0,0] >;`

C9⋊C54 in GAP, Magma, Sage, TeX

`C_9\rtimes C_{54}`
`% in TeX`

`G:=Group("C9:C54");`
`// GroupNames label`

`G:=SmallGroup(486,30);`
`// by ID`

`G=gap.SmallGroup(486,30);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,8104,3250,208,11669]);`
`// Polycyclic`

`G:=Group<a,b|a^9=b^54=1,b*a*b^-1=a^2>;`
`// generators/relations`

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