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## G = C12⋊S4order 288 = 25·32

### 1st semidirect product of C12 and S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×A4 — C12⋊S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C2×C3⋊S4 — C12⋊S4
 Lower central C3×A4 — C6×A4 — C12⋊S4
 Upper central C1 — C2 — C4

Generators and relations for C12⋊S4
G = < a,b,c,d,e | a12=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: 1028 in 144 conjugacy classes, 27 normal (16 characteristic)
C1, C2, C2 [×4], C3, C3 [×3], C4, C4 [×3], C22, C22 [×8], S3 [×8], C6, C6 [×5], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], C12, C12 [×4], A4 [×3], D6 [×12], C2×C6, C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3⋊S3 [×2], C3×C6, D12 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], S4 [×6], C2×A4 [×3], C22×S3 [×2], C22×C6, C4⋊D4, C3×C12, C3×A4, C2×C3⋊S3 [×2], C4⋊Dic3, D6⋊C4 [×2], C4×A4 [×3], C2×D12, C2×C3⋊D4 [×2], C22×C12, C2×S4 [×6], C12⋊S3, C3⋊S4 [×2], C6×A4, C127D4, C4⋊S4 [×3], C12×A4, C2×C3⋊S4 [×2], C12⋊S4
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], C3⋊S3, D12 [×4], S4, C2×C3⋊S3, C2×S4, C12⋊S3, C3⋊S4, C4⋊S4, C2×C3⋊S4, C12⋊S4

Character table of C12⋊S4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J size 1 1 3 3 36 36 2 8 8 8 2 6 36 36 2 6 6 8 8 8 2 2 6 6 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 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 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 1 1 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 1 1 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 2 -1 -1 -1 -2 -2 0 0 2 2 2 -1 -1 -1 -2 -2 -2 -2 1 1 1 1 1 1 orthogonal lifted from D6 ρ6 2 2 2 2 0 0 -1 -1 2 -1 -2 -2 0 0 -1 -1 -1 2 -1 -1 1 1 1 1 -2 1 -2 1 1 1 orthogonal lifted from D6 ρ7 2 2 2 2 0 0 -1 2 -1 -1 2 2 0 0 -1 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 2 -1 -1 2 -1 orthogonal lifted from S3 ρ8 2 2 2 2 0 0 -1 -1 -1 2 2 2 0 0 -1 -1 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 -1 2 -1 2 orthogonal lifted from S3 ρ9 2 -2 2 -2 0 0 2 2 2 2 0 0 0 0 -2 -2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -1 -1 2 -1 2 2 0 0 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 2 0 0 -1 2 -1 -1 -2 -2 0 0 -1 -1 -1 -1 2 -1 1 1 1 1 1 -2 1 1 -2 1 orthogonal lifted from D6 ρ12 2 2 2 2 0 0 2 -1 -1 -1 2 2 0 0 2 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 2 0 0 -1 -1 -1 2 -2 -2 0 0 -1 -1 -1 -1 -1 2 1 1 1 1 1 1 1 -2 1 -2 orthogonal lifted from D6 ρ14 2 -2 2 -2 0 0 -1 -1 2 -1 0 0 0 0 1 1 -1 -2 1 1 √3 -√3 √3 -√3 0 -√3 0 √3 √3 -√3 orthogonal lifted from D12 ρ15 2 -2 2 -2 0 0 2 -1 -1 -1 0 0 0 0 -2 -2 2 1 1 1 0 0 0 0 √3 -√3 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ16 2 -2 2 -2 0 0 -1 -1 -1 2 0 0 0 0 1 1 -1 1 1 -2 -√3 √3 -√3 √3 -√3 -√3 √3 0 √3 0 orthogonal lifted from D12 ρ17 2 -2 2 -2 0 0 -1 -1 2 -1 0 0 0 0 1 1 -1 -2 1 1 -√3 √3 -√3 √3 0 √3 0 -√3 -√3 √3 orthogonal lifted from D12 ρ18 2 -2 2 -2 0 0 -1 2 -1 -1 0 0 0 0 1 1 -1 1 -2 1 -√3 √3 -√3 √3 √3 0 -√3 √3 0 -√3 orthogonal lifted from D12 ρ19 2 -2 2 -2 0 0 2 -1 -1 -1 0 0 0 0 -2 -2 2 1 1 1 0 0 0 0 -√3 √3 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ20 2 -2 2 -2 0 0 -1 2 -1 -1 0 0 0 0 1 1 -1 1 -2 1 √3 -√3 √3 -√3 -√3 0 √3 -√3 0 √3 orthogonal lifted from D12 ρ21 2 -2 2 -2 0 0 -1 -1 -1 2 0 0 0 0 1 1 -1 1 1 -2 √3 -√3 √3 -√3 √3 √3 -√3 0 -√3 0 orthogonal lifted from D12 ρ22 3 3 -1 -1 -1 -1 3 0 0 0 3 -1 1 1 3 -1 -1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ23 3 3 -1 -1 1 1 3 0 0 0 3 -1 -1 -1 3 -1 -1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ24 3 3 -1 -1 -1 1 3 0 0 0 -3 1 1 -1 3 -1 -1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ25 3 3 -1 -1 1 -1 3 0 0 0 -3 1 -1 1 3 -1 -1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ26 6 6 -2 -2 0 0 -3 0 0 0 6 -2 0 0 -3 1 1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C3⋊S4 ρ27 6 -6 -2 2 0 0 6 0 0 0 0 0 0 0 -6 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4 ρ28 6 6 -2 -2 0 0 -3 0 0 0 -6 2 0 0 -3 1 1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×C3⋊S4 ρ29 6 -6 -2 2 0 0 -3 0 0 0 0 0 0 0 3 -1 1 0 0 0 -3√3 3√3 √3 -√3 0 0 0 0 0 0 orthogonal faithful ρ30 6 -6 -2 2 0 0 -3 0 0 0 0 0 0 0 3 -1 1 0 0 0 3√3 -3√3 -√3 √3 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of C12⋊S4
On 36 points
Generators in S36
```(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 28 29 30 31 32 33 34 35 36)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 14 27)(2 15 28)(3 16 29)(4 17 30)(5 18 31)(6 19 32)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 25)(12 13 26)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 28)(14 27)(15 26)(16 25)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)```

`G:=sub<Sym(36)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,14,27)(2,15,28)(3,16,29)(4,17,30)(5,18,31)(6,19,32)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,25)(12,13,26), (2,12)(3,11)(4,10)(5,9)(6,8)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)>;`

`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)(25,26,27,28,29,30,31,32,33,34,35,36), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,14,27)(2,15,28)(3,16,29)(4,17,30)(5,18,31)(6,19,32)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,25)(12,13,26), (2,12)(3,11)(4,10)(5,9)(6,8)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29) );`

`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),(25,26,27,28,29,30,31,32,33,34,35,36)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,14,27),(2,15,28),(3,16,29),(4,17,30),(5,18,31),(6,19,32),(7,20,33),(8,21,34),(9,22,35),(10,23,36),(11,24,25),(12,13,26)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,28),(14,27),(15,26),(16,25),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29)])`

Matrix representation of C12⋊S4 in GL5(𝔽13)

 8 7 0 0 0 12 9 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 12 0 0 0 0 12 0 1 0 0 12 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 1 0 12 0 0 0 0 12
,
 4 2 0 0 0 9 8 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 12 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(5,GF(13))| [8,12,0,0,0,7,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[4,9,0,0,0,2,8,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;`

C12⋊S4 in GAP, Magma, Sage, TeX

`C_{12}\rtimes S_4`
`% in TeX`

`G:=Group("C12:S4");`
`// GroupNames label`

`G:=SmallGroup(288,909);`
`// by ID`

`G=gap.SmallGroup(288,909);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,451,1684,6053,782,3534,1350]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=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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