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## G = C27⋊C6order 162 = 2·34

### The semidirect product of C27 and C6 acting faithfully

Aliases: C27⋊C6, D27⋊C3, C32.D9, C27⋊C3⋊C2, C9.3(C3×S3), (C3×C9).3S3, C3.3(C3×D9), SmallGroup(162,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C27⋊C6
 Chief series C1 — C3 — C9 — C27 — C27⋊C3 — C27⋊C6
 Lower central C27 — C27⋊C6
 Upper central C1

Generators and relations for C27⋊C6
G = < a,b | a27=b6=1, bab-1=a17 >

Character table of C27⋊C6

 class 1 2 3A 3B 3C 6A 6B 9A 9B 9C 9D 9E 27A 27B 27C 27D 27E 27F 27G 27H 27I size 1 27 2 3 3 27 27 2 2 2 6 6 6 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 ζ32 ζ3 ζ6 ζ65 1 1 1 ζ3 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 1 linear of order 6 ρ4 1 -1 1 ζ3 ζ32 ζ65 ζ6 1 1 1 ζ32 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 1 linear of order 6 ρ5 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ3 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 1 linear of order 3 ρ6 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ32 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 1 linear of order 3 ρ7 2 0 2 2 2 0 0 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 2 2 0 0 -1 -1 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ9 2 0 2 2 2 0 0 -1 -1 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ10 2 0 2 2 2 0 0 -1 -1 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 0 2 -1-√-3 -1+√-3 0 0 2 2 2 -1+√-3 -1-√-3 -1 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 -1 -1 complex lifted from C3×S3 ρ12 2 0 2 -1+√-3 -1-√-3 0 0 2 2 2 -1-√-3 -1+√-3 -1 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 -1 -1 complex lifted from C3×S3 ρ13 2 0 2 -1+√-3 -1-√-3 0 0 -1 -1 -1 ζ6 ζ65 ζ95+ζ94 ζ97+ζ95 ζ98+ζ94 ζ94+ζ92 ζ95+ζ9 ζ98+ζ97 ζ92+ζ9 ζ98+ζ9 ζ97+ζ92 complex lifted from C3×D9 ρ14 2 0 2 -1+√-3 -1-√-3 0 0 -1 -1 -1 ζ6 ζ65 ζ97+ζ92 ζ92+ζ9 ζ97+ζ95 ζ98+ζ97 ζ94+ζ92 ζ95+ζ9 ζ98+ζ94 ζ95+ζ94 ζ98+ζ9 complex lifted from C3×D9 ρ15 2 0 2 -1-√-3 -1+√-3 0 0 -1 -1 -1 ζ65 ζ6 ζ97+ζ92 ζ98+ζ97 ζ94+ζ92 ζ92+ζ9 ζ97+ζ95 ζ98+ζ94 ζ95+ζ9 ζ95+ζ94 ζ98+ζ9 complex lifted from C3×D9 ρ16 2 0 2 -1-√-3 -1+√-3 0 0 -1 -1 -1 ζ65 ζ6 ζ95+ζ94 ζ94+ζ92 ζ95+ζ9 ζ97+ζ95 ζ98+ζ94 ζ92+ζ9 ζ98+ζ97 ζ98+ζ9 ζ97+ζ92 complex lifted from C3×D9 ρ17 2 0 2 -1-√-3 -1+√-3 0 0 -1 -1 -1 ζ65 ζ6 ζ98+ζ9 ζ95+ζ9 ζ98+ζ97 ζ98+ζ94 ζ92+ζ9 ζ97+ζ95 ζ94+ζ92 ζ97+ζ92 ζ95+ζ94 complex lifted from C3×D9 ρ18 2 0 2 -1+√-3 -1-√-3 0 0 -1 -1 -1 ζ6 ζ65 ζ98+ζ9 ζ98+ζ94 ζ92+ζ9 ζ95+ζ9 ζ98+ζ97 ζ94+ζ92 ζ97+ζ95 ζ97+ζ92 ζ95+ζ94 complex lifted from C3×D9 ρ19 6 0 -3 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ20 6 0 -3 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ21 6 0 -3 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C27⋊C6
On 27 points - transitive group 27T55
Generators in S27
```(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)
(2 9 11 27 20 18)(3 17 21 26 12 8)(4 25)(5 6 14 24 23 15)(7 22)(10 19)(13 16)```

`G:=sub<Sym(27)| (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), (2,9,11,27,20,18)(3,17,21,26,12,8)(4,25)(5,6,14,24,23,15)(7,22)(10,19)(13,16)>;`

`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), (2,9,11,27,20,18)(3,17,21,26,12,8)(4,25)(5,6,14,24,23,15)(7,22)(10,19)(13,16) );`

`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)], [(2,9,11,27,20,18),(3,17,21,26,12,8),(4,25),(5,6,14,24,23,15),(7,22),(10,19),(13,16)])`

`G:=TransitiveGroup(27,55);`

C27⋊C6 is a maximal subgroup of   C27⋊C18  C33.D9  He3.3D9  He3.4D9  C33.5D9  He3.5D9
C27⋊C6 is a maximal quotient of   C27⋊C12  C273C18  C32⋊D27  C33.5D9

Matrix representation of C27⋊C6 in GL6(𝔽109)

 0 0 59 82 0 0 0 0 27 32 0 0 0 0 0 0 59 82 0 0 0 0 27 32 1 0 0 0 0 0 0 1 0 0 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 32 50 0 0 0 0 82 77 0 0 50 27 0 0 0 0 77 59 0 0

`G:=sub<GL(6,GF(109))| [0,0,0,0,1,0,0,0,0,0,0,1,59,27,0,0,0,0,82,32,0,0,0,0,0,0,59,27,0,0,0,0,82,32,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,50,77,0,0,0,0,27,59,0,0,32,82,0,0,0,0,50,77,0,0] >;`

C27⋊C6 in GAP, Magma, Sage, TeX

`C_{27}\rtimes C_6`
`% in TeX`

`G:=Group("C27:C6");`
`// GroupNames label`

`G:=SmallGroup(162,9);`
`// by ID`

`G=gap.SmallGroup(162,9);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-3,-3,452,457,237,1803,138,2704]);`
`// Polycyclic`

`G:=Group<a,b|a^27=b^6=1,b*a*b^-1=a^17>;`
`// generators/relations`

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