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G = C4.4S3≀C2order 288 = 25·32

4th non-split extension by C4 of S3≀C2 acting via S3≀C2/C32⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — C4.4S3≀C2
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — S32⋊C4 — C4.4S3≀C2
 Lower central C32 — C2×C3⋊S3 — C4.4S3≀C2
 Upper central C1 — C2 — C4

Generators and relations for C4.4S3≀C2
G = < a,b,c,d,e | a4=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=a2d-1 >

Subgroups: 696 in 122 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×5], C22 [×7], S3 [×6], C6 [×4], C2×C4 [×5], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×4], C12 [×4], D6 [×8], C2×C6 [×2], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3⋊S3 [×2], C3×C6, Dic6 [×3], C4×S3 [×4], D12, C3⋊D4 [×2], C3×D4, C3×Q8, C22×S3 [×2], C4.4D4, C3×Dic3 [×2], C3⋊Dic3, C3×C12, C32⋊C4 [×2], S32 [×4], S3×C6 [×2], C2×C3⋊S3, S3×D4, S3×Q8, C6.D6 [×2], D6⋊S3, C322Q8, C3×Dic6, C3×D12, C4×C3⋊S3, C2×C32⋊C4 [×2], C2×S32 [×2], S32⋊C4 [×4], C4×C32⋊C4, Dic3.D6, D6⋊D6, C4.4S3≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D4 [×2], C4.4D4, S3≀C2, C2×S3≀C2, C4.4S3≀C2

Character table of C4.4S3≀C2

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 12A 12B 12C 12D size 1 1 9 9 12 12 4 4 2 12 12 18 18 18 18 18 4 4 24 24 8 8 24 24 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 -1 1 1 1 1 -1 1 -1 -1 -1 1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 -2 -2 0 0 2 2 -2 0 0 2 0 0 0 0 2 2 0 0 -2 -2 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 2 2 2 0 0 -2 0 0 0 0 2 2 0 0 2 2 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 2 2 0 0 0 0 -2i 2i 0 0 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ12 2 -2 -2 2 0 0 2 2 0 0 0 0 0 0 2i -2i -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ13 2 -2 2 -2 0 0 2 2 0 0 0 0 2i -2i 0 0 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 -2 2 0 0 2 2 0 0 0 0 0 0 -2i 2i -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 4 4 0 0 -2 2 -2 1 -4 0 0 0 0 0 0 0 1 -2 1 -1 -1 2 0 0 orthogonal lifted from C2×S3≀C2 ρ16 4 4 0 0 0 0 1 -2 -4 -2 2 0 0 0 0 0 -2 1 0 0 2 -1 1 -1 orthogonal lifted from C2×S3≀C2 ρ17 4 4 0 0 2 -2 -2 1 -4 0 0 0 0 0 0 0 1 -2 -1 1 -1 2 0 0 orthogonal lifted from C2×S3≀C2 ρ18 4 4 0 0 0 0 1 -2 4 2 2 0 0 0 0 0 -2 1 0 0 -2 1 -1 -1 orthogonal lifted from S3≀C2 ρ19 4 4 0 0 -2 -2 -2 1 4 0 0 0 0 0 0 0 1 -2 1 1 1 -2 0 0 orthogonal lifted from S3≀C2 ρ20 4 4 0 0 0 0 1 -2 4 -2 -2 0 0 0 0 0 -2 1 0 0 -2 1 1 1 orthogonal lifted from S3≀C2 ρ21 4 4 0 0 2 2 -2 1 4 0 0 0 0 0 0 0 1 -2 -1 -1 1 -2 0 0 orthogonal lifted from S3≀C2 ρ22 4 4 0 0 0 0 1 -2 -4 2 -2 0 0 0 0 0 -2 1 0 0 2 -1 -1 1 orthogonal lifted from C2×S3≀C2 ρ23 8 -8 0 0 0 0 -4 2 0 0 0 0 0 0 0 0 -2 4 0 0 0 0 0 0 orthogonal faithful ρ24 8 -8 0 0 0 0 2 -4 0 0 0 0 0 0 0 0 4 -2 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C4.4S3≀C2
On 24 points - transitive group 24T642
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)
(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)
(1 21)(2 22)(3 23)(4 24)(5 14 10 19)(6 15 11 20)(7 16 12 17)(8 13 9 18)
(1 4)(2 3)(5 11)(6 10)(7 9)(8 12)(13 14)(15 16)(17 20)(18 19)(21 22)(23 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), (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,14,19)(2,15,20)(3,16,17)(4,13,18), (1,21)(2,22)(3,23)(4,24)(5,14,10,19)(6,15,11,20)(7,16,12,17)(8,13,9,18), (1,4)(2,3)(5,11)(6,10)(7,9)(8,12)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)>;`

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

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

`G:=TransitiveGroup(24,642);`

Matrix representation of C4.4S3≀C2 in GL6(𝔽13)

 0 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C4.4S3≀C2 in GAP, Magma, Sage, TeX

`C_4._4S_3\wr C_2`
`% in TeX`

`G:=Group("C4.4S3wrC2");`
`// GroupNames label`

`G:=SmallGroup(288,869);`
`// by ID`

`G=gap.SmallGroup(288,869);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,422,219,100,2693,2028,362,797,1203]);`
`// Polycyclic`

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

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