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## G = C3⋊S3.5D8order 288 = 25·32

### The non-split extension by C3⋊S3 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3⋊S3.5D8
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C4⋊(C32⋊C4) — C3⋊S3.5D8
 Lower central C32 — C3×C6 — C3×C12 — C3⋊S3.5D8
 Upper central C1 — C2 — C4 — D4

Generators and relations for C3⋊S3.5D8
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=c, ab=ba, cac=a-1, dad-1=eae-1=ab-1, cbc=b-1, dbd-1=ebe-1=a-1b-1, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 720 in 104 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×2], C22 [×5], S3 [×8], C6 [×6], C8, C2×C4 [×2], D4, D4 [×2], C23, C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×4], C4⋊C4, C2×C8, C2×D4, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], D4⋊C4, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3 [×3], C62, S3×D4 [×2], C322C8, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C2×C32⋊C4, C22×C3⋊S3, C3⋊S33C8, C4⋊(C32⋊C4), D4×C3⋊S3, C3⋊S3.5D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, D4⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C3⋊S3.5D8

Character table of C3⋊S3.5D8

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B size 1 1 4 9 9 36 4 4 2 18 36 36 4 4 8 8 8 8 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 i -i 1 1 1 1 1 1 i -i -i i 1 1 linear of order 4 ρ6 1 1 -1 -1 -1 1 1 1 1 -1 i -i 1 1 -1 -1 -1 -1 -i i i -i 1 1 linear of order 4 ρ7 1 1 -1 -1 -1 1 1 1 1 -1 -i i 1 1 -1 -1 -1 -1 i -i -i i 1 1 linear of order 4 ρ8 1 1 1 -1 -1 -1 1 1 1 -1 -i i 1 1 1 1 1 1 -i i i -i 1 1 linear of order 4 ρ9 2 2 0 2 2 0 2 2 -2 -2 0 0 2 2 0 0 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 0 -2 -2 0 2 2 -2 2 0 0 2 2 0 0 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ11 2 -2 0 -2 2 0 2 2 0 0 0 0 -2 -2 0 0 0 0 -√2 √2 -√2 √2 0 0 orthogonal lifted from D8 ρ12 2 -2 0 -2 2 0 2 2 0 0 0 0 -2 -2 0 0 0 0 √2 -√2 √2 -√2 0 0 orthogonal lifted from D8 ρ13 2 -2 0 2 -2 0 2 2 0 0 0 0 -2 -2 0 0 0 0 √-2 √-2 -√-2 -√-2 0 0 complex lifted from SD16 ρ14 2 -2 0 2 -2 0 2 2 0 0 0 0 -2 -2 0 0 0 0 -√-2 -√-2 √-2 √-2 0 0 complex lifted from SD16 ρ15 4 4 -4 0 0 0 1 -2 4 0 0 0 -2 1 2 -1 -1 2 0 0 0 0 -2 1 orthogonal lifted from C2×C32⋊C4 ρ16 4 4 0 0 0 0 -2 1 -4 0 0 0 1 -2 3 0 0 -3 0 0 0 0 -1 2 orthogonal lifted from C62⋊C4 ρ17 4 4 0 0 0 0 -2 1 -4 0 0 0 1 -2 -3 0 0 3 0 0 0 0 -1 2 orthogonal lifted from C62⋊C4 ρ18 4 4 4 0 0 0 1 -2 4 0 0 0 -2 1 -2 1 1 -2 0 0 0 0 -2 1 orthogonal lifted from C32⋊C4 ρ19 4 4 0 0 0 0 1 -2 -4 0 0 0 -2 1 0 3 -3 0 0 0 0 0 2 -1 orthogonal lifted from C62⋊C4 ρ20 4 4 -4 0 0 0 -2 1 4 0 0 0 1 -2 -1 2 2 -1 0 0 0 0 1 -2 orthogonal lifted from C2×C32⋊C4 ρ21 4 4 4 0 0 0 -2 1 4 0 0 0 1 -2 1 -2 -2 1 0 0 0 0 1 -2 orthogonal lifted from C32⋊C4 ρ22 4 4 0 0 0 0 1 -2 -4 0 0 0 -2 1 0 -3 3 0 0 0 0 0 2 -1 orthogonal lifted from C62⋊C4 ρ23 8 -8 0 0 0 0 2 -4 0 0 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ24 8 -8 0 0 0 0 -4 2 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C3⋊S3.5D8
On 24 points - transitive group 24T676
Generators in S24
```(1 10 21)(3 23 12)(5 14 17)(7 19 16)
(1 10 21)(2 11 22)(3 23 12)(4 24 13)(5 14 17)(6 15 18)(7 19 16)(8 20 9)
(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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 10 20 21)(11 16 22 19)(12 18 23 15)(13 14 24 17)```

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

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

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

`G:=TransitiveGroup(24,676);`

On 24 points - transitive group 24T677
Generators in S24
```(1 13 22)(2 14 23)(3 24 15)(4 17 16)(5 9 18)(6 10 19)(7 20 11)(8 21 12)
(2 23 14)(4 16 17)(6 19 10)(8 12 21)
(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 21 22 16)(10 15 23 20)(11 19 24 14)(12 13 17 18)```

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

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

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

`G:=TransitiveGroup(24,677);`

Matrix representation of C3⋊S3.5D8 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 0 1 0 0 0 0 72 72
,
 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 1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 72 72
,
 61 67 0 0 0 0 12 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 71 72 0 0 49 49 0 0 0 0 49 48 0 0
,
 0 6 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 72 0 0 0 0 2 1 0 0 49 49 0 0 0 0 49 48 0 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[61,12,0,0,0,0,67,0,0,0,0,0,0,0,0,0,49,49,0,0,0,0,49,48,0,0,72,71,0,0,0,0,1,72,0,0],[0,12,0,0,0,0,6,0,0,0,0,0,0,0,0,0,49,49,0,0,0,0,49,48,0,0,1,2,0,0,0,0,72,1,0,0] >;`

C3⋊S3.5D8 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3._5D_8`
`% in TeX`

`G:=Group("C3:S3.5D8");`
`// GroupNames label`

`G:=SmallGroup(288,430);`
`// by ID`

`G=gap.SmallGroup(288,430);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,675,346,80,9413,691,12550,2372]);`
`// 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=a^-1,d*a*d^-1=e*a*e^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^-1*b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;`
`// generators/relations`

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