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## G = C3⋊S3order 18 = 2·32

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

Aliases: C3⋊S3, C322C2, SmallGroup(18,4)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3⋊S3
G = < a,b,c | a3=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Character table of C3⋊S3

 class 1 2 3A 3B 3C 3D size 1 9 2 2 2 2 ρ1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 linear of order 2 ρ3 2 0 -1 -1 -1 2 orthogonal lifted from S3 ρ4 2 0 2 -1 -1 -1 orthogonal lifted from S3 ρ5 2 0 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 0 -1 -1 2 -1 orthogonal lifted from S3

Permutation representations of C3⋊S3
On 9 points - transitive group 9T5
Generators in S9
```(1 2 3)(4 5 6)(7 8 9)
(1 5 8)(2 6 9)(3 4 7)
(2 3)(4 9)(5 8)(6 7)```

`G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (1,5,8)(2,6,9)(3,4,7), (2,3)(4,9)(5,8)(6,7)>;`

`G:=Group( (1,2,3)(4,5,6)(7,8,9), (1,5,8)(2,6,9)(3,4,7), (2,3)(4,9)(5,8)(6,7) );`

`G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9)], [(1,5,8),(2,6,9),(3,4,7)], [(2,3),(4,9),(5,8),(6,7)])`

`G:=TransitiveGroup(9,5);`

Regular action on 18 points - transitive group 18T4
Generators in S18
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 8)(2 6 9)(3 4 7)(10 16 13)(11 17 14)(12 18 15)
(1 10)(2 12)(3 11)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)```

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

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

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

`G:=TransitiveGroup(18,4);`

C3⋊S3 is a maximal subgroup of
C32⋊C4  C32⋊C6  C33⋊C2  C3⋊S4  C52⋊(C3⋊S3)
C3⋊D3p: S32  C9⋊S3  C3⋊D15  C3⋊D21  C3⋊D33  C3⋊D39  C3⋊D51  C3⋊D57 ...
C3⋊S3 is a maximal quotient of
C3⋊Dic3  He3⋊C2  C33⋊C2  C3⋊S4  C52⋊(C3⋊S3)
C3⋊D3p: C9⋊S3  C3⋊D15  C3⋊D21  C3⋊D33  C3⋊D39  C3⋊D51  C3⋊D57  C3⋊D69 ...

Polynomial with Galois group C3⋊S3 over ℚ
actionf(x)Disc(f)
9T5x9-54x7-12x6+756x5+180x4-2652x3-864x2+288x+64232·318·72·238·432·1512

Matrix representation of C3⋊S3 in GL4(ℤ) generated by

 0 1 0 0 -1 -1 0 0 0 0 0 1 0 0 -1 -1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 -1
,
 1 0 0 0 -1 -1 0 0 0 0 0 -1 0 0 -1 0
`G:=sub<GL(4,Integers())| [0,-1,0,0,1,-1,0,0,0,0,0,-1,0,0,1,-1],[1,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,-1],[1,-1,0,0,0,-1,0,0,0,0,0,-1,0,0,-1,0] >;`

C3⋊S3 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3`
`% in TeX`

`G:=Group("C3:S3");`
`// GroupNames label`

`G:=SmallGroup(18,4);`
`// by ID`

`G=gap.SmallGroup(18,4);`
`# by ID`

`G:=PCGroup([3,-2,-3,-3,25,110]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=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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