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## G = C8⋊D6order 96 = 25·3

### 1st semidirect product of C8 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C8⋊D6
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C8⋊D6
 Lower central C3 — C6 — C12 — C8⋊D6
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C8⋊D6
G = < a,b,c | a8=b6=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 210 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C24, Dic6, C4×S3, D12, D12, D12, C3⋊D4, C2×C12, C22×S3, C8⋊C22, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, C8⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C8⋊C22, C2×D12, C8⋊D6

Character table of C8⋊D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 8A 8B 12A 12B 12C 24A 24B 24C 24D size 1 1 2 12 12 12 2 2 2 12 2 4 4 4 2 2 4 4 4 4 4 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 0 0 0 2 2 -2 0 2 -2 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 0 -1 2 2 0 -1 -1 -2 -2 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 0 0 0 -1 2 2 0 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 0 0 0 -1 -2 2 0 -1 1 -2 2 -1 -1 1 1 -1 -1 1 orthogonal lifted from D6 ρ13 2 2 2 0 0 0 2 -2 -2 0 2 2 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 0 -1 -2 2 0 -1 1 2 -2 -1 -1 1 -1 1 1 -1 orthogonal lifted from D6 ρ15 2 2 -2 0 0 0 -1 2 -2 0 -1 1 0 0 1 1 -1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ16 2 2 -2 0 0 0 -1 2 -2 0 -1 1 0 0 1 1 -1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ17 2 2 2 0 0 0 -1 -2 -2 0 -1 -1 0 0 1 1 1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ18 2 2 2 0 0 0 -1 -2 -2 0 -1 -1 0 0 1 1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ19 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 -2√3 2√3 0 0 0 0 0 orthogonal faithful ρ21 4 -4 0 0 0 0 -2 0 0 0 2 0 0 0 2√3 -2√3 0 0 0 0 0 orthogonal faithful

Permutation representations of C8⋊D6
On 24 points - transitive group 24T107
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)
(1 13 21)(2 10 22 6 14 18)(3 15 23)(4 12 24 8 16 20)(5 9 17)(7 11 19)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 24)(7 23)(8 22)(10 16)(11 15)(12 14)```

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

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

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

`G:=TransitiveGroup(24,107);`

Matrix representation of C8⋊D6 in GL4(𝔽73) generated by

 0 0 1 0 0 0 0 1 7 14 0 0 59 66 0 0
,
 1 1 0 0 72 0 0 0 0 0 72 72 0 0 1 0
,
 72 72 0 0 0 1 0 0 0 0 66 7 0 0 14 7
`G:=sub<GL(4,GF(73))| [0,0,7,59,0,0,14,66,1,0,0,0,0,1,0,0],[1,72,0,0,1,0,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,72,1,0,0,0,0,66,14,0,0,7,7] >;`

C8⋊D6 in GAP, Magma, Sage, TeX

`C_8\rtimes D_6`
`% in TeX`

`G:=Group("C8:D6");`
`// GroupNames label`

`G:=SmallGroup(96,115);`
`// by ID`

`G=gap.SmallGroup(96,115);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,188,50,579,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^6=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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