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## G = C8⋊S4order 192 = 26·3

### 3rd semidirect product of C8 and S4 acting via S4/A4=C2

Aliases: C83S4, A41M4(2), A4⋊C4.C4, (C2×S4).C4, A4⋊C84C2, (C8×A4)⋊5C2, C2.7(C4×S4), (C4×S4).2C2, C4.27(C2×S4), (C22×C8)⋊4S3, C22⋊(C8⋊S3), C23.2(C4×S3), (C22×C4).9D6, (C4×A4).13C22, (C2×A4).2(C2×C4), SmallGroup(192,959)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C8⋊S4
 Chief series C1 — C22 — A4 — C2×A4 — C4×A4 — C4×S4 — C8⋊S4
 Lower central A4 — C2×A4 — C8⋊S4
 Upper central C1 — C4 — C8

Generators and relations for C8⋊S4
G = < a,b,c,d,e | a8=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a5, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 246 in 72 conjugacy classes, 17 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×4], C22, C22 [×5], S3, C6, C8, C8 [×3], C2×C4 [×7], D4 [×2], C23, C23, Dic3, C12, A4, D6, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, S4, C2×A4, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C8⋊S3, A4⋊C4, C4×A4, C2×S4, C89D4, A4⋊C8, C8×A4, C4×S4, C8⋊S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, M4(2), C4×S3, S4, C8⋊S3, C2×S4, C4×S4, C8⋊S4

Character table of C8⋊S4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 6 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 24A 24B 24C 24D size 1 1 3 3 12 8 1 1 3 3 12 12 12 8 2 2 6 6 12 12 12 12 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 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 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 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 linear of order 2 ρ5 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 i -i i -i i -i i -i -1 -1 i -i i -i linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -i i -i i i -i i -i -1 -1 -i i -i i linear of order 4 ρ7 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 i -i i -i -i i -i i -1 -1 i -i i -i linear of order 4 ρ8 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -i i -i i -i i -i i -1 -1 -i i -i i linear of order 4 ρ9 2 2 2 2 0 -1 2 2 2 2 0 0 0 -1 -2 -2 -2 -2 0 0 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 0 -1 2 2 2 2 0 0 0 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 -2 -2 2 0 2 2i -2i 2i -2i 0 0 0 -2 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from M4(2) ρ12 2 2 2 2 0 -1 -2 -2 -2 -2 0 0 0 -1 -2i 2i -2i 2i 0 0 0 0 1 1 i -i i -i complex lifted from C4×S3 ρ13 2 2 2 2 0 -1 -2 -2 -2 -2 0 0 0 -1 2i -2i 2i -2i 0 0 0 0 1 1 -i i -i i complex lifted from C4×S3 ρ14 2 -2 -2 2 0 2 -2i 2i -2i 2i 0 0 0 -2 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 complex lifted from M4(2) ρ15 2 -2 -2 2 0 -1 -2i 2i -2i 2i 0 0 0 1 0 0 0 0 0 0 0 0 -i i 2ζ8ζ3+ζ8 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 2ζ87ζ3+ζ87 complex lifted from C8⋊S3 ρ16 2 -2 -2 2 0 -1 2i -2i 2i -2i 0 0 0 1 0 0 0 0 0 0 0 0 i -i 2ζ87ζ3+ζ87 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 2ζ8ζ3+ζ8 complex lifted from C8⋊S3 ρ17 2 -2 -2 2 0 -1 -2i 2i -2i 2i 0 0 0 1 0 0 0 0 0 0 0 0 -i i 2ζ85ζ3+ζ85 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 2ζ83ζ3+ζ83 complex lifted from C8⋊S3 ρ18 2 -2 -2 2 0 -1 2i -2i 2i -2i 0 0 0 1 0 0 0 0 0 0 0 0 i -i 2ζ83ζ3+ζ83 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 2ζ85ζ3+ζ85 complex lifted from C8⋊S3 ρ19 3 3 -1 -1 -1 0 3 3 -1 -1 1 1 -1 0 -3 -3 1 1 -1 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ20 3 3 -1 -1 1 0 3 3 -1 -1 -1 -1 1 0 -3 -3 1 1 1 1 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ21 3 3 -1 -1 1 0 3 3 -1 -1 -1 -1 1 0 3 3 -1 -1 -1 -1 1 1 0 0 0 0 0 0 orthogonal lifted from S4 ρ22 3 3 -1 -1 -1 0 3 3 -1 -1 1 1 -1 0 3 3 -1 -1 1 1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ23 3 3 -1 -1 -1 0 -3 -3 1 1 -1 1 1 0 -3i 3i i -i i -i -i i 0 0 0 0 0 0 complex lifted from C4×S4 ρ24 3 3 -1 -1 -1 0 -3 -3 1 1 -1 1 1 0 3i -3i -i i -i i i -i 0 0 0 0 0 0 complex lifted from C4×S4 ρ25 3 3 -1 -1 1 0 -3 -3 1 1 1 -1 -1 0 -3i 3i i -i -i i i -i 0 0 0 0 0 0 complex lifted from C4×S4 ρ26 3 3 -1 -1 1 0 -3 -3 1 1 1 -1 -1 0 3i -3i -i i i -i -i i 0 0 0 0 0 0 complex lifted from C4×S4 ρ27 6 -6 2 -2 0 0 -6i 6i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ28 6 -6 2 -2 0 0 6i -6i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C8⋊S4
On 24 points - transitive group 24T323
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 13)(10 14)(11 15)(12 16)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 11 19)(2 12 20)(3 13 21)(4 14 22)(5 15 23)(6 16 24)(7 9 17)(8 10 18)
(2 6)(4 8)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(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), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,11,19)(2,12,20)(3,13,21)(4,14,22)(5,15,23)(6,16,24)(7,9,17)(8,10,18), (2,6)(4,8)(9,17)(10,22)(11,19)(12,24)(13,21)(14,18)(15,23)(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), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,11,19)(2,12,20)(3,13,21)(4,14,22)(5,15,23)(6,16,24)(7,9,17)(8,10,18), (2,6)(4,8)(9,17)(10,22)(11,19)(12,24)(13,21)(14,18)(15,23)(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)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24)], [(1,11,19),(2,12,20),(3,13,21),(4,14,22),(5,15,23),(6,16,24),(7,9,17),(8,10,18)], [(2,6),(4,8),(9,17),(10,22),(11,19),(12,24),(13,21),(14,18),(15,23),(16,20)])`

`G:=TransitiveGroup(24,323);`

Matrix representation of C8⋊S4 in GL5(𝔽73)

 45 13 0 0 0 26 28 0 0 0 0 0 46 0 0 0 0 0 46 0 0 0 0 0 46
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 72 72 72 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 72 72 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 72 72 72 0 0 0 0 1
,
 72 54 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(5,GF(73))| [45,26,0,0,0,13,28,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,72,0,0,0,1,72,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,1,0,72,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,72,1],[72,0,0,0,0,54,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;`

C8⋊S4 in GAP, Magma, Sage, TeX

`C_8\rtimes S_4`
`% in TeX`

`G:=Group("C8:S4");`
`// GroupNames label`

`G:=SmallGroup(192,959);`
`// by ID`

`G=gap.SmallGroup(192,959);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,141,36,58,1124,4037,285,2358,475]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^8=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^5,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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