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## G = C3.S4order 72 = 23·32

### The non-split extension by C3 of S4 acting via S4/A4=C2

Aliases: C3.S4, C22⋊D9, C3.A4⋊C2, (C2×C6).S3, SmallGroup(72,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3.A4 — C3.S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C3.S4
 Lower central C3.A4 — C3.S4
 Upper central C1

Generators and relations for C3.S4
G = < a,b,c,d,e | a3=b2=c2=e2=1, d3=a, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=a-1d2 >

3C2
18C2
9C4
9C22
3C6
6S3
4C9
9D4
3D6
3Dic3
4D9

Character table of C3.S4

 class 1 2A 2B 3 4 6 9A 9B 9C size 1 3 18 2 18 6 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 1 1 1 1 linear of order 2 ρ3 2 2 0 2 0 2 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 -1 0 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ5 2 2 0 -1 0 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ6 2 2 0 -1 0 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ7 3 -1 -1 3 1 -1 0 0 0 orthogonal lifted from S4 ρ8 3 -1 1 3 -1 -1 0 0 0 orthogonal lifted from S4 ρ9 6 -2 0 -3 0 1 0 0 0 orthogonal faithful

Permutation representations of C3.S4
On 18 points - transitive group 18T38
Generators in S18
```(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 15)(2 16)(4 18)(5 10)(7 12)(8 13)
(2 16)(3 17)(5 10)(6 11)(8 13)(9 14)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(2 9)(3 8)(4 7)(5 6)(10 11)(12 18)(13 17)(14 16)```

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

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

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

`G:=TransitiveGroup(18,38);`

On 18 points - transitive group 18T39
Generators in S18
```(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 12)(2 13)(4 15)(5 16)(7 18)(8 10)
(2 13)(3 14)(5 16)(6 17)(8 10)(9 11)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 12)(2 11)(3 10)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)```

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

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

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

`G:=TransitiveGroup(18,39);`

C3.S4 is a maximal subgroup of
C32.S4  C9⋊S4  C32.3S4  C42⋊D9  C24⋊D9  C22⋊D45
C3.S4 is a maximal quotient of
Q8.D9  Q8⋊D9  C6.S4  C9.S4  C32.3S4  C42⋊D9  C24⋊D9  C22⋊D45

Matrix representation of C3.S4 in GL5(𝔽37)

 36 1 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 36
,
 26 31 0 0 0 6 20 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

`G:=sub<GL(5,GF(37))| [36,36,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[26,6,0,0,0,31,20,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;`

C3.S4 in GAP, Magma, Sage, TeX

`C_3.S_4`
`% in TeX`

`G:=Group("C3.S4");`
`// GroupNames label`

`G:=SmallGroup(72,15);`
`// by ID`

`G=gap.SmallGroup(72,15);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-2,2,101,66,182,723,368,454,684]);`
`// Polycyclic`

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

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