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## G = C27⋊3C18order 486 = 2·35

### The semidirect product of C27 and C18 acting via C18/C3=C6

Aliases: C273C18, D272C9, C92.1S3, C272C9⋊C2, (C3×C27).C6, C9.4(S3×C9), (C3×D27).C3, C3.3(C9×D9), (C3×C9).1D9, C3.2(C27⋊C6), C32.13(C3×D9), (C3×C9).53(C3×S3), SmallGroup(486,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C27⋊3C18
 Chief series C1 — C3 — C9 — C27 — C3×C27 — C27⋊2C9 — C27⋊3C18
 Lower central C27 — C27⋊3C18
 Upper central C1 — C3

Generators and relations for C273C18
G = < a,b | a27=b18=1, bab-1=a17 >

Smallest permutation representation of C273C18
On 54 points
Generators in S54
```(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)
(1 32 13 47 25 35 10 50 22 38 7 53 19 41 4 29 16 44)(2 40 23 46 17 52 11 31 5 37 26 43 20 49 14 28 8 34)(3 48 6 45 9 42 12 39 15 36 18 33 21 30 24 54 27 51)```

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

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

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

63 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 9A ··· 9I 9J ··· 9O 9P ··· 9U 18A ··· 18F 27A ··· 27AA order 1 2 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 27 ··· 27 size 1 27 1 1 2 2 2 27 27 2 ··· 2 3 ··· 3 6 ··· 6 27 ··· 27 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 6 6 type + + + + + image C1 C2 C3 C6 C9 C18 S3 D9 C3×S3 S3×C9 C3×D9 C9×D9 C27⋊C6 C27⋊3C18 kernel C27⋊3C18 C27⋊2C9 C3×D27 C3×C27 D27 C27 C92 C3×C9 C3×C9 C9 C32 C3 C3 C1 # reps 1 1 2 2 6 6 1 3 2 6 6 18 3 6

Matrix representation of C273C18 in GL6(𝔽109)

 100 98 0 0 0 0 64 9 27 0 0 0 99 15 0 0 0 0 0 0 0 60 0 44 0 0 0 40 0 73 0 0 0 14 105 49
,
 0 0 0 66 0 0 0 0 0 87 27 0 0 0 0 21 0 16 66 0 0 0 0 0 87 27 0 0 0 0 21 0 16 0 0 0

`G:=sub<GL(6,GF(109))| [100,64,99,0,0,0,98,9,15,0,0,0,0,27,0,0,0,0,0,0,0,60,40,14,0,0,0,0,0,105,0,0,0,44,73,49],[0,0,0,66,87,21,0,0,0,0,27,0,0,0,0,0,0,16,66,87,21,0,0,0,0,27,0,0,0,0,0,0,16,0,0,0] >;`

C273C18 in GAP, Magma, Sage, TeX

`C_{27}\rtimes_3C_{18}`
`% in TeX`

`G:=Group("C27:3C18");`
`// GroupNames label`

`G:=SmallGroup(486,15);`
`// by ID`

`G=gap.SmallGroup(486,15);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,2163,2169,381,8104,208,11669]);`
`// Polycyclic`

`G:=Group<a,b|a^27=b^18=1,b*a*b^-1=a^17>;`
`// generators/relations`

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