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## G = C4⋊S4order 96 = 25·3

### The semidirect product of C4 and S4 acting via S4/A4=C2

Aliases: C4⋊S4, A41D4, C22⋊D12, C23.3D6, (C2×S4)⋊1C2, (C4×A4)⋊1C2, C2.4(C2×S4), (C22×C4)⋊2S3, (C2×A4).3C22, SmallGroup(96,187)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C4⋊S4
 Chief series C1 — C22 — A4 — C2×A4 — C2×S4 — C4⋊S4
 Lower central A4 — C2×A4 — C4⋊S4
 Upper central C1 — C2 — C4

Generators and relations for C4⋊S4
G = < a,b,c,d,e | a4=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 218 in 56 conjugacy classes, 12 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, D4, C23, C23, C12, A4, D6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D12, S4, C2×A4, C4⋊D4, C4×A4, C2×S4, C4⋊S4
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2×S4, C4⋊S4

Character table of C4⋊S4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6 12A 12B size 1 1 3 3 12 12 8 2 6 12 12 8 8 8 ρ1 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 linear of order 2 ρ3 1 1 1 1 -1 1 1 -1 -1 1 -1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ5 2 2 2 2 0 0 -1 2 2 0 0 -1 -1 -1 orthogonal lifted from S3 ρ6 2 -2 2 -2 0 0 2 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ7 2 2 2 2 0 0 -1 -2 -2 0 0 -1 1 1 orthogonal lifted from D6 ρ8 2 -2 2 -2 0 0 -1 0 0 0 0 1 -√3 √3 orthogonal lifted from D12 ρ9 2 -2 2 -2 0 0 -1 0 0 0 0 1 √3 -√3 orthogonal lifted from D12 ρ10 3 3 -1 -1 1 1 0 3 -1 -1 -1 0 0 0 orthogonal lifted from S4 ρ11 3 3 -1 -1 -1 -1 0 3 -1 1 1 0 0 0 orthogonal lifted from S4 ρ12 3 3 -1 -1 1 -1 0 -3 1 1 -1 0 0 0 orthogonal lifted from C2×S4 ρ13 3 3 -1 -1 -1 1 0 -3 1 -1 1 0 0 0 orthogonal lifted from C2×S4 ρ14 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C4⋊S4
On 12 points - transitive group 12T54
Generators in S12
```(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(9 11)(10 12)
(5 7)(6 8)(9 11)(10 12)
(1 9 7)(2 10 8)(3 11 5)(4 12 6)
(1 4)(2 3)(5 10)(6 9)(7 12)(8 11)```

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

`G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(9,11)(10,12), (5,7)(6,8)(9,11)(10,12), (1,9,7)(2,10,8)(3,11,5)(4,12,6), (1,4)(2,3)(5,10)(6,9)(7,12)(8,11) );`

`G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,3),(2,4),(9,11),(10,12)], [(5,7),(6,8),(9,11),(10,12)], [(1,9,7),(2,10,8),(3,11,5),(4,12,6)], [(1,4),(2,3),(5,10),(6,9),(7,12),(8,11)]])`

`G:=TransitiveGroup(12,54);`

On 16 points - transitive group 16T191
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6)(2 7)(3 8)(4 5)(9 13)(10 14)(11 15)(12 16)
(1 16)(2 13)(3 14)(4 15)(5 11)(6 12)(7 9)(8 10)
(5 15 11)(6 16 12)(7 13 9)(8 14 10)
(1 4)(2 3)(5 12)(6 11)(7 10)(8 9)(13 14)(15 16)```

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

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

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

`G:=TransitiveGroup(16,191);`

On 24 points - transitive group 24T128
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 3)(2 4)(5 7)(6 8)(9 13)(10 14)(11 15)(12 16)(17 23)(18 24)(19 21)(20 22)
(1 23 15)(2 24 16)(3 21 13)(4 22 14)(5 17 9)(6 18 10)(7 19 11)(8 20 12)
(2 4)(5 7)(9 19)(10 18)(11 17)(12 20)(13 21)(14 24)(15 23)(16 22)```

`G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,3)(2,4)(5,7)(6,8)(9,13)(10,14)(11,15)(12,16)(17,23)(18,24)(19,21)(20,22), (1,23,15)(2,24,16)(3,21,13)(4,22,14)(5,17,9)(6,18,10)(7,19,11)(8,20,12), (2,4)(5,7)(9,19)(10,18)(11,17)(12,20)(13,21)(14,24)(15,23)(16,22)>;`

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

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

`G:=TransitiveGroup(24,128);`

On 24 points - transitive group 24T170
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(1 11 16)(2 12 13)(3 9 14)(4 10 15)(5 19 21)(6 20 22)(7 17 23)(8 18 24)
(1 21)(2 24)(3 23)(4 22)(5 16)(6 15)(7 14)(8 13)(9 17)(10 20)(11 19)(12 18)```

`G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (1,11,16)(2,12,13)(3,9,14)(4,10,15)(5,19,21)(6,20,22)(7,17,23)(8,18,24), (1,21)(2,24)(3,23)(4,22)(5,16)(6,15)(7,14)(8,13)(9,17)(10,20)(11,19)(12,18)>;`

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

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

`G:=TransitiveGroup(24,170);`

On 24 points - transitive group 24T171
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 6)(2 7)(3 8)(4 5)(9 14)(10 15)(11 16)(12 13)
(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 13 17)(6 14 18)(7 15 19)(8 16 20)
(2 4)(5 7)(9 23)(10 22)(11 21)(12 24)(13 19)(14 18)(15 17)(16 20)```

`G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13), (9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,10,24)(3,11,21)(4,12,22)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (2,4)(5,7)(9,23)(10,22)(11,21)(12,24)(13,19)(14,18)(15,17)(16,20)>;`

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

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

`G:=TransitiveGroup(24,171);`

On 24 points - transitive group 24T172
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 6)(2 7)(3 8)(4 5)(9 14)(10 15)(11 16)(12 13)
(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 13 17)(6 14 18)(7 15 19)(8 16 20)
(1 5)(2 8)(3 7)(4 6)(9 17)(10 20)(11 19)(12 18)(13 23)(14 22)(15 21)(16 24)```

`G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13), (9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,10,24)(3,11,21)(4,12,22)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (1,5)(2,8)(3,7)(4,6)(9,17)(10,20)(11,19)(12,18)(13,23)(14,22)(15,21)(16,24)>;`

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

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

`G:=TransitiveGroup(24,172);`

C4⋊S4 is a maximal subgroup of
C82S4  A4⋊D8  D4⋊S4  Q83S4  C24.10D6  D4×S4  Q84S4  Dic3⋊S4  C12⋊S4  C4⋊S5  Dic5⋊S4  C20⋊S4
C4⋊S4 is a maximal quotient of
Q8.D12  Q8⋊D12  Q8.2D12  A4⋊Q16  C82S4  A4⋊D8  C8.S4  C8.4S4  C8.3S4  C24.4D6  C24.5D6  C22⋊D36  Dic3⋊S4  C12⋊S4  Dic5⋊S4  C20⋊S4

Polynomial with Galois group C4⋊S4 over ℚ
actionf(x)Disc(f)
12T54x12-3x10+2x8-x6-2x4-3x2-1-228·296

Matrix representation of C4⋊S4 in GL5(ℤ)

 -1 -1 0 0 0 2 1 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 0 -1 1 0 0 0 -1 0 0 0 1 -1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 -1 0 1 0 0 -1 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 1 -1 0 0 0 0 -1 1
,
 -1 -1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 -1 0 0 0 1 0 -1

`G:=sub<GL(5,Integers())| [-1,2,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,-1,-1,-1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,-1,-1,-1,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,-1,-1,-1,0,0,0,0,1],[-1,0,0,0,0,-1,1,0,0,0,0,0,1,1,1,0,0,0,-1,0,0,0,0,0,-1] >;`

C4⋊S4 in GAP, Magma, Sage, TeX

`C_4\rtimes S_4`
`% in TeX`

`G:=Group("C4:S4");`
`// GroupNames label`

`G:=SmallGroup(96,187);`
`// by ID`

`G=gap.SmallGroup(96,187);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-2,2,73,31,387,1444,202,869,347]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^3=e^2=1,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=d^-1>;`
`// generators/relations`

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