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## G = C9⋊S3order 54 = 2·33

### The semidirect product of C9 and S3 acting via S3/C3=C2

Aliases: C9⋊S3, C3⋊D9, C32.3S3, (C3×C9)⋊3C2, C3.(C3⋊S3), SmallGroup(54,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C9⋊S3
 Lower central C3×C9 — C9⋊S3
 Upper central C1

Generators and relations for C9⋊S3
G = < a,b,c | a9=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Character table of C9⋊S3

 class 1 2 3A 3B 3C 3D 9A 9B 9C 9D 9E 9F 9G 9H 9I size 1 27 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 2 0 -1 -1 -1 2 -1 -1 2 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 0 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ5 2 0 -1 -1 -1 2 -1 -1 -1 -1 -1 2 2 2 -1 orthogonal lifted from S3 ρ6 2 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ7 2 0 -1 -1 2 -1 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ8 2 0 -1 -1 2 -1 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ9 2 0 2 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ10 2 0 -1 2 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 0 -1 2 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 orthogonal lifted from D9 ρ12 2 0 2 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ13 2 0 -1 -1 2 -1 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ14 2 0 2 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ15 2 0 -1 2 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 orthogonal lifted from D9

Permutation representations of C9⋊S3
On 27 points - transitive group 27T10
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)
(1 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 25)(6 16 26)(7 17 27)(8 18 19)(9 10 20)
(2 9)(3 8)(4 7)(5 6)(10 22)(11 21)(12 20)(13 19)(14 27)(15 26)(16 25)(17 24)(18 23)```

`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), (1,11,21)(2,12,22)(3,13,23)(4,14,24)(5,15,25)(6,16,26)(7,17,27)(8,18,19)(9,10,20), (2,9)(3,8)(4,7)(5,6)(10,22)(11,21)(12,20)(13,19)(14,27)(15,26)(16,25)(17,24)(18,23)>;`

`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), (1,11,21)(2,12,22)(3,13,23)(4,14,24)(5,15,25)(6,16,26)(7,17,27)(8,18,19)(9,10,20), (2,9)(3,8)(4,7)(5,6)(10,22)(11,21)(12,20)(13,19)(14,27)(15,26)(16,25)(17,24)(18,23) );`

`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)], [(1,11,21),(2,12,22),(3,13,23),(4,14,24),(5,15,25),(6,16,26),(7,17,27),(8,18,19),(9,10,20)], [(2,9),(3,8),(4,7),(5,6),(10,22),(11,21),(12,20),(13,19),(14,27),(15,26),(16,25),(17,24),(18,23)]])`

`G:=TransitiveGroup(27,10);`

C9⋊S3 is a maximal subgroup of
S3×D9  C32⋊D9  He3.S3  He3.2S3  C9⋊D9  C27⋊S3  C33.S3  He3.4S3  C324D9  C9⋊S4  C32.3S4  C3⋊D45  C3⋊D63
C9⋊S3 is a maximal quotient of
C9⋊Dic3  C9⋊D9  C322D9  C27⋊S3  C324D9  C9⋊S4  C32.3S4  C3⋊D45  C3⋊D63

Matrix representation of C9⋊S3 in GL4(𝔽19) generated by

 12 17 0 0 2 14 0 0 0 0 2 14 0 0 5 7
,
 1 0 0 0 0 1 0 0 0 0 18 18 0 0 1 0
,
 1 0 0 0 18 18 0 0 0 0 18 18 0 0 0 1
`G:=sub<GL(4,GF(19))| [12,2,0,0,17,14,0,0,0,0,2,5,0,0,14,7],[1,0,0,0,0,1,0,0,0,0,18,1,0,0,18,0],[1,18,0,0,0,18,0,0,0,0,18,0,0,0,18,1] >;`

C9⋊S3 in GAP, Magma, Sage, TeX

`C_9\rtimes S_3`
`% in TeX`

`G:=Group("C9:S3");`
`// GroupNames label`

`G:=SmallGroup(54,7);`
`// by ID`

`G=gap.SmallGroup(54,7);`
`# by ID`

`G:=PCGroup([4,-2,-3,-3,-3,177,149,146,579]);`
`// Polycyclic`

`G:=Group<a,b,c|a^9=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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