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## G = C3⋊S3.2D8order 288 = 25·32

### 1st non-split extension by C3⋊S3 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C3⋊S3.2D8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C4×C3⋊S3 — D6⋊D6 — C3⋊S3.2D8
 Lower central C32 — C3×C6 — C3⋊Dic3 — C3⋊S3.2D8
 Upper central C1 — C2 — C4

Generators and relations for C3⋊S3.2D8
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=c, ab=ba, cac=dbd-1=a-1, dad-1=eae-1=cbc=b-1, ebe-1=a, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 560 in 88 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×2], C22 [×5], S3 [×6], C6 [×4], C8, C2×C4 [×2], D4 [×3], C23, C32, Dic3 [×2], C12 [×2], D6 [×8], C2×C6 [×2], C4⋊C4, C2×C8, C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8, C24, C4×S3 [×2], D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], D4⋊C4, C3⋊Dic3, C3×C12, C32⋊C4, S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×C8, S3×D4, C3×C3⋊C8, D6⋊S3 [×2], C3×D12, C4×C3⋊S3, C2×C32⋊C4, C2×S32, C12.29D6, C4⋊(C32⋊C4), D6⋊D6, C3⋊S3.2D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, D4⋊C4, S3≀C2, S32⋊C4, C3⋊S3.2D8

Character table of C3⋊S3.2D8

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 24A 24B 24C 24D size 1 1 9 9 12 12 4 4 2 18 36 36 4 4 24 24 6 6 6 6 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 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 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 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 linear of order 2 ρ5 1 1 -1 -1 1 -1 1 1 -1 1 -i i 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 -i i 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 i -i 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 i -i 1 1 -1 1 i -i i -i -1 -1 -1 -i -i i i linear of order 4 ρ9 2 2 -2 -2 0 0 2 2 2 -2 0 0 2 2 0 0 0 0 0 0 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 2 -2 -2 0 0 2 2 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 -2 0 0 √2 -√2 -√2 √2 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ12 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 -2 0 0 -√2 √2 √2 -√2 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 √-2 √-2 -√-2 -√-2 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ14 2 -2 -2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 -√-2 -√-2 √-2 √-2 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ15 4 4 0 0 0 0 1 -2 4 0 0 0 -2 1 0 0 -2 -2 -2 -2 1 1 -2 1 1 1 1 orthogonal lifted from S3≀C2 ρ16 4 4 0 0 2 -2 -2 1 -4 0 0 0 1 -2 -1 1 0 0 0 0 2 2 -1 0 0 0 0 orthogonal lifted from S32⋊C4 ρ17 4 4 0 0 2 2 -2 1 4 0 0 0 1 -2 -1 -1 0 0 0 0 -2 -2 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ18 4 4 0 0 -2 -2 -2 1 4 0 0 0 1 -2 1 1 0 0 0 0 -2 -2 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ19 4 4 0 0 0 0 1 -2 4 0 0 0 -2 1 0 0 2 2 2 2 1 1 -2 -1 -1 -1 -1 orthogonal lifted from S3≀C2 ρ20 4 4 0 0 -2 2 -2 1 -4 0 0 0 1 -2 1 -1 0 0 0 0 2 2 -1 0 0 0 0 orthogonal lifted from S32⋊C4 ρ21 4 4 0 0 0 0 1 -2 -4 0 0 0 -2 1 0 0 -2i 2i -2i 2i -1 -1 2 -i -i i i complex lifted from S32⋊C4 ρ22 4 4 0 0 0 0 1 -2 -4 0 0 0 -2 1 0 0 2i -2i 2i -2i -1 -1 2 i i -i -i complex lifted from S32⋊C4 ρ23 4 -4 0 0 0 0 1 -2 0 0 0 0 2 -1 0 0 2ζ87 2ζ85 2ζ83 2ζ8 3i -3i 0 ζ8 ζ85 ζ87 ζ83 complex faithful ρ24 4 -4 0 0 0 0 1 -2 0 0 0 0 2 -1 0 0 2ζ8 2ζ83 2ζ85 2ζ87 -3i 3i 0 ζ87 ζ83 ζ8 ζ85 complex faithful ρ25 4 -4 0 0 0 0 1 -2 0 0 0 0 2 -1 0 0 2ζ83 2ζ8 2ζ87 2ζ85 3i -3i 0 ζ85 ζ8 ζ83 ζ87 complex faithful ρ26 4 -4 0 0 0 0 1 -2 0 0 0 0 2 -1 0 0 2ζ85 2ζ87 2ζ8 2ζ83 -3i 3i 0 ζ83 ζ87 ζ85 ζ8 complex faithful ρ27 8 -8 0 0 0 0 -4 2 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C3⋊S3.2D8
On 24 points - transitive group 24T666
Generators in S24
```(2 22 10)(4 24 12)(6 18 14)(8 20 16)
(1 9 21)(3 11 23)(5 13 17)(7 15 19)
(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)
(1 4 5 8)(2 7 6 3)(9 12 17 20)(10 19 18 11)(13 16 21 24)(14 23 22 15)```

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

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

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

`G:=TransitiveGroup(24,666);`

On 24 points - transitive group 24T667
Generators in S24
```(1 21 9)(3 23 11)(5 17 13)(7 19 15)
(2 10 22)(4 12 24)(6 14 18)(8 16 20)
(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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 20 21 16)(10 15 22 19)(11 18 23 14)(12 13 24 17)```

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

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

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

`G:=TransitiveGroup(24,667);`

Matrix representation of C3⋊S3.2D8 in GL6(𝔽73)

 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 0 1 0 0 0 0 72 72
,
 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 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72
,
 6 6 0 0 0 0 67 6 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
,
 6 6 0 0 0 0 6 67 0 0 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 72 0 0 0 0 0 0 72 0 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[6,67,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,1,0,0,0,0,0,1,0,0] >;`

C3⋊S3.2D8 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3._2D_8`
`% in TeX`

`G:=Group("C3:S3.2D8");`
`// GroupNames label`

`G:=SmallGroup(288,377);`
`// by ID`

`G=gap.SmallGroup(288,377);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,422,100,675,80,2693,2028,691,797,2372]);`
`// Polycyclic`

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

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