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## G = C4.S3≀C2order 288 = 25·32

### 1st non-split extension by C4 of S3≀C2 acting via S3≀C2/S32=C2

Aliases: C4.9S3≀C2, (C3×C12).1D4, C3⋊Dic3.2D4, C32⋊(C4.D4), D6⋊D6.1C2, C12.31D67C2, C32⋊M4(2)⋊1C2, (C2×S32).C4, C2.3(S32⋊C4), (C4×C3⋊S3).1C22, (C3×C6).2(C22⋊C4), (C2×C3⋊S3).6(C2×C4), SmallGroup(288,375)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — C4.S3≀C2
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — D6⋊D6 — C4.S3≀C2
 Lower central C32 — C3×C6 — C2×C3⋊S3 — C4.S3≀C2
 Upper central C1 — C2 — C4

Generators and relations for C4.S3≀C2
G = < a,b,c,d,e | a4=b3=c3=e2=1, d4=a2, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=a-1d3 >

Subgroups: 520 in 83 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2 [×3], C3 [×2], C4, C4, C22 [×5], S3 [×5], C6 [×4], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, Dic3 [×2], C12 [×2], D6 [×8], C2×C6 [×2], M4(2) [×2], C2×D4, C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3 [×2], D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C4.D4, C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, C8⋊S3, S3×D4, C3×C3⋊C8, C322C8, D6⋊S3, C3×D12, C4×C3⋊S3, C2×S32 [×2], C12.31D6, C32⋊M4(2), D6⋊D6, C4.S3≀C2
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C4.D4, S3≀C2, S32⋊C4, C4.S3≀C2

Character table of C4.S3≀C2

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 24A 24B 24C 24D size 1 1 12 12 18 4 4 2 18 4 4 24 24 12 12 36 36 4 4 8 12 12 12 12 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 1 -i i -i i -1 -1 -1 -i -i i i linear of order 4 ρ6 1 1 -1 1 1 1 1 -1 -1 1 1 1 -1 -i i i -i -1 -1 -1 -i -i i i linear of order 4 ρ7 1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 i -i i -i -1 -1 -1 i i -i -i linear of order 4 ρ8 1 1 -1 1 1 1 1 -1 -1 1 1 1 -1 i -i -i i -1 -1 -1 i i -i -i linear of order 4 ρ9 2 2 0 0 -2 2 2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 -2 2 2 2 -2 2 2 0 0 0 0 0 0 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -2 -2 0 -2 1 4 0 -2 1 1 1 0 0 0 0 -2 -2 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ12 4 -4 0 0 0 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ13 4 4 2 -2 0 -2 1 -4 0 -2 1 1 -1 0 0 0 0 2 2 -1 0 0 0 0 orthogonal lifted from S32⋊C4 ρ14 4 4 2 2 0 -2 1 4 0 -2 1 -1 -1 0 0 0 0 -2 -2 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ15 4 4 0 0 0 1 -2 4 0 1 -2 0 0 2 2 0 0 1 1 -2 -1 -1 -1 -1 orthogonal lifted from S3≀C2 ρ16 4 4 -2 2 0 -2 1 -4 0 -2 1 -1 1 0 0 0 0 2 2 -1 0 0 0 0 orthogonal lifted from S32⋊C4 ρ17 4 4 0 0 0 1 -2 4 0 1 -2 0 0 -2 -2 0 0 1 1 -2 1 1 1 1 orthogonal lifted from S3≀C2 ρ18 4 4 0 0 0 1 -2 -4 0 1 -2 0 0 -2i 2i 0 0 -1 -1 2 i i -i -i complex lifted from S32⋊C4 ρ19 4 4 0 0 0 1 -2 -4 0 1 -2 0 0 2i -2i 0 0 -1 -1 2 -i -i i i complex lifted from S32⋊C4 ρ20 4 -4 0 0 0 1 -2 0 0 -1 2 0 0 0 0 0 0 3i -3i 0 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 complex faithful ρ21 4 -4 0 0 0 1 -2 0 0 -1 2 0 0 0 0 0 0 3i -3i 0 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 complex faithful ρ22 4 -4 0 0 0 1 -2 0 0 -1 2 0 0 0 0 0 0 -3i 3i 0 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 complex faithful ρ23 4 -4 0 0 0 1 -2 0 0 -1 2 0 0 0 0 0 0 -3i 3i 0 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 complex faithful ρ24 8 -8 0 0 0 -4 2 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C4.S3≀C2
On 24 points - transitive group 24T662
Generators in S24
```(1 7 5 3)(2 4 6 8)(9 18 13 22)(10 23 14 19)(11 20 15 24)(12 17 16 21)
(1 17 10)(3 12 19)(5 21 14)(7 16 23)
(2 11 18)(4 20 13)(6 15 22)(8 24 9)
(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 3)(2 8)(4 6)(5 7)(9 18)(10 12)(11 24)(13 22)(14 16)(15 20)(17 19)(21 23)```

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

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

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

`G:=TransitiveGroup(24,662);`

Matrix representation of C4.S3≀C2 in GL4(𝔽5) generated by

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

C4.S3≀C2 in GAP, Magma, Sage, TeX

`C_4.S_3\wr C_2`
`% in TeX`

`G:=Group("C4.S3wrC2");`
`// GroupNames label`

`G:=SmallGroup(288,375);`
`// by ID`

`G=gap.SmallGroup(288,375);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,219,675,80,2693,2028,691,797,2372]);`
`// Polycyclic`

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

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