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## G = C3⋊S3⋊3C8order 144 = 24·32

### 2nd semidirect product of C3⋊S3 and C8 acting via C8/C4=C2

Aliases: C3⋊S33C8, C322(C2×C8), (C3×C12).2C4, C322C85C2, C4.3(C32⋊C4), C3⋊Dic3.7C22, (C4×C3⋊S3).6C2, (C2×C3⋊S3).5C4, (C3×C6).1(C2×C4), C2.1(C2×C32⋊C4), SmallGroup(144,130)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3⋊3C8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C3⋊S3⋊3C8
 Lower central C32 — C3⋊S3⋊3C8
 Upper central C1 — C4

Generators and relations for C3⋊S33C8
G = < a,b,c,d | a3=b3=c2=d8=1, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a-1b-1, cd=dc >

Character table of C3⋊S33C8

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D size 1 1 9 9 4 4 1 1 9 9 4 4 9 9 9 9 9 9 9 9 4 4 4 4 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 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 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 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 linear of order 2 ρ5 1 1 -1 -1 1 1 1 1 -1 -1 1 1 i -i i i -i -i i -i 1 1 1 1 linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 1 1 i i -i -i -i -i i i -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -i i -i -i i i -i i 1 1 1 1 linear of order 4 ρ8 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -i -i i i i i -i -i -1 -1 -1 -1 linear of order 4 ρ9 1 -1 1 -1 1 1 i -i i -i -1 -1 ζ8 ζ85 ζ83 ζ87 ζ83 ζ87 ζ85 ζ8 -i -i i i linear of order 8 ρ10 1 -1 -1 1 1 1 i -i -i i -1 -1 ζ87 ζ87 ζ8 ζ85 ζ85 ζ8 ζ83 ζ83 -i -i i i linear of order 8 ρ11 1 -1 -1 1 1 1 -i i i -i -1 -1 ζ8 ζ8 ζ87 ζ83 ζ83 ζ87 ζ85 ζ85 i i -i -i linear of order 8 ρ12 1 -1 1 -1 1 1 -i i -i i -1 -1 ζ87 ζ83 ζ85 ζ8 ζ85 ζ8 ζ83 ζ87 i i -i -i linear of order 8 ρ13 1 -1 -1 1 1 1 i -i -i i -1 -1 ζ83 ζ83 ζ85 ζ8 ζ8 ζ85 ζ87 ζ87 -i -i i i linear of order 8 ρ14 1 -1 -1 1 1 1 -i i i -i -1 -1 ζ85 ζ85 ζ83 ζ87 ζ87 ζ83 ζ8 ζ8 i i -i -i linear of order 8 ρ15 1 -1 1 -1 1 1 i -i i -i -1 -1 ζ85 ζ8 ζ87 ζ83 ζ87 ζ83 ζ8 ζ85 -i -i i i linear of order 8 ρ16 1 -1 1 -1 1 1 -i i -i i -1 -1 ζ83 ζ87 ζ8 ζ85 ζ8 ζ85 ζ87 ζ83 i i -i -i linear of order 8 ρ17 4 4 0 0 -2 1 4 4 0 0 -2 1 0 0 0 0 0 0 0 0 -2 1 1 -2 orthogonal lifted from C32⋊C4 ρ18 4 4 0 0 -2 1 -4 -4 0 0 -2 1 0 0 0 0 0 0 0 0 2 -1 -1 2 orthogonal lifted from C2×C32⋊C4 ρ19 4 4 0 0 1 -2 4 4 0 0 1 -2 0 0 0 0 0 0 0 0 1 -2 -2 1 orthogonal lifted from C32⋊C4 ρ20 4 4 0 0 1 -2 -4 -4 0 0 1 -2 0 0 0 0 0 0 0 0 -1 2 2 -1 orthogonal lifted from C2×C32⋊C4 ρ21 4 -4 0 0 1 -2 -4i 4i 0 0 -1 2 0 0 0 0 0 0 0 0 i -2i 2i -i complex faithful ρ22 4 -4 0 0 1 -2 4i -4i 0 0 -1 2 0 0 0 0 0 0 0 0 -i 2i -2i i complex faithful ρ23 4 -4 0 0 -2 1 -4i 4i 0 0 2 -1 0 0 0 0 0 0 0 0 -2i i -i 2i complex faithful ρ24 4 -4 0 0 -2 1 4i -4i 0 0 2 -1 0 0 0 0 0 0 0 0 2i -i i -2i complex faithful

Permutation representations of C3⋊S33C8
On 24 points - transitive group 24T244
Generators in S24
```(1 21 9)(2 22 10)(3 11 23)(4 12 24)(5 17 13)(6 18 14)(7 15 19)(8 16 20)
(2 10 22)(4 24 12)(6 14 18)(8 20 16)
(1 5)(2 6)(3 7)(4 8)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 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)```

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

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

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

`G:=TransitiveGroup(24,244);`

Matrix representation of C3⋊S33C8 in GL4(𝔽5) generated by

 2 1 3 0 1 2 1 4 4 0 0 2 4 3 1 4
,
 0 0 4 3 3 0 0 0 0 3 0 3 0 0 0 1
,
 2 4 2 3 1 3 4 0 0 2 4 3 1 2 4 1
,
 1 0 0 2 3 0 0 4 0 0 3 1 0 1 3 1
`G:=sub<GL(4,GF(5))| [2,1,4,4,1,2,0,3,3,1,0,1,0,4,2,4],[0,3,0,0,0,0,3,0,4,0,0,0,3,0,3,1],[2,1,0,1,4,3,2,2,2,4,4,4,3,0,3,1],[1,3,0,0,0,0,0,1,0,0,3,3,2,4,1,1] >;`

C3⋊S33C8 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3\rtimes_3C_8`
`% in TeX`

`G:=Group("C3:S3:3C8");`
`// GroupNames label`

`G:=SmallGroup(144,130);`
`// by ID`

`G=gap.SmallGroup(144,130);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,55,50,3364,256,4613,881]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^2=d^8=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,c*d=d*c>;`
`// generators/relations`

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