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## G = C3⋊S3⋊D8order 288 = 25·32

### The semidirect product of C3⋊S3 and D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C3⋊S3⋊D8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8 — C3⋊S3⋊D8
 Lower central C32 — C3×C6 — C3⋊Dic3 — C3⋊S3⋊D8
 Upper central C1 — C2 — C4

Generators and relations for C3⋊S3⋊D8
G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, cac=dbd-1=a-1, dad-1=b, ae=ea, cbc=ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 848 in 130 conjugacy classes, 25 normal (11 characteristic)
C1, C2, C2 [×6], C3 [×2], C4, C4, C22 [×9], S3 [×8], C6 [×6], C8 [×2], C2×C4, D4 [×6], C23 [×2], C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×4], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], C2×D8, C3⋊Dic3, C3×C12, S32 [×4], S3×C6 [×4], C2×C3⋊S3, S3×D4 [×2], C322C8 [×2], D6⋊S3 [×4], C3×D12 [×2], C4×C3⋊S3, C2×S32 [×2], C32⋊D8 [×4], C3⋊S33C8, D6⋊D6 [×2], C3⋊S3⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, C2×D8, S3≀C2, C2×S3≀C2, C3⋊S3⋊D8

Character table of C3⋊S3⋊D8

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B size 1 1 9 9 12 12 12 12 4 4 2 18 4 4 24 24 24 24 18 18 18 18 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 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 0 0 2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 0 0 2 2 2 -2 2 2 0 0 0 0 0 0 0 0 2 2 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 √2 -√2 √2 -√2 0 0 orthogonal lifted from D8 ρ12 2 -2 -2 2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 -√2 -√2 √2 √2 0 0 orthogonal lifted from D8 ρ13 2 -2 2 -2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 -√2 √2 -√2 √2 0 0 orthogonal lifted from D8 ρ14 2 -2 -2 2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 √2 √2 -√2 -√2 0 0 orthogonal lifted from D8 ρ15 4 4 0 0 -2 2 0 0 1 -2 -4 0 1 -2 -1 1 0 0 0 0 0 0 -1 2 orthogonal lifted from C2×S3≀C2 ρ16 4 4 0 0 2 -2 0 0 1 -2 -4 0 1 -2 1 -1 0 0 0 0 0 0 -1 2 orthogonal lifted from C2×S3≀C2 ρ17 4 4 0 0 0 0 2 -2 -2 1 -4 0 -2 1 0 0 -1 1 0 0 0 0 2 -1 orthogonal lifted from C2×S3≀C2 ρ18 4 4 0 0 0 0 2 2 -2 1 4 0 -2 1 0 0 -1 -1 0 0 0 0 -2 1 orthogonal lifted from S3≀C2 ρ19 4 4 0 0 -2 -2 0 0 1 -2 4 0 1 -2 1 1 0 0 0 0 0 0 1 -2 orthogonal lifted from S3≀C2 ρ20 4 4 0 0 2 2 0 0 1 -2 4 0 1 -2 -1 -1 0 0 0 0 0 0 1 -2 orthogonal lifted from S3≀C2 ρ21 4 4 0 0 0 0 -2 -2 -2 1 4 0 -2 1 0 0 1 1 0 0 0 0 -2 1 orthogonal lifted from S3≀C2 ρ22 4 4 0 0 0 0 -2 2 -2 1 -4 0 -2 1 0 0 1 -1 0 0 0 0 2 -1 orthogonal lifted from C2×S3≀C2 ρ23 8 -8 0 0 0 0 0 0 2 -4 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ24 8 -8 0 0 0 0 0 0 -4 2 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C3⋊S3⋊D8
On 24 points - transitive group 24T659
Generators in S24
```(1 11 21)(2 12 22)(3 23 13)(4 24 14)(5 15 17)(6 16 18)(7 19 9)(8 20 10)
(1 11 21)(2 22 12)(3 23 13)(4 14 24)(5 15 17)(6 18 16)(7 19 9)(8 10 20)
(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 8)(2 7)(3 6)(4 5)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 24)(16 23)```

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

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

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

`G:=TransitiveGroup(24,659);`

Matrix representation of C3⋊S3⋊D8 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 57 16 0 0 0 0 57 57 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 72 0 0 0
,
 57 16 0 0 0 0 16 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,1,0,0],[57,16,0,0,0,0,16,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C3⋊S3⋊D8 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3\rtimes D_8`
`% in TeX`

`G:=Group("C3:S3:D8");`
`// GroupNames label`

`G:=SmallGroup(288,873);`
`// by ID`

`G=gap.SmallGroup(288,873);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,100,675,346,80,2693,2028,362,797,1203]);`
`// Polycyclic`

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

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