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## G = C8.D6order 96 = 25·3

### 1st non-split extension by C8 of 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 — C4○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=1, b6=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b5 >

Subgroups: 146 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C24, Dic6, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, 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 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 12A 12B 12C 24A 24B 24C 24D size 1 1 2 12 2 2 2 12 12 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 -1 2 2 0 0 0 -1 -1 -2 -2 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 2 0 -1 2 2 0 0 0 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 0 2 -2 -2 0 0 0 2 2 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 -1 -2 2 0 0 0 -1 1 2 -2 -1 -1 1 1 -1 -1 1 orthogonal lifted from D6 ρ13 2 2 -2 0 -1 -2 2 0 0 0 -1 1 -2 2 -1 -1 1 -1 1 1 -1 orthogonal lifted from D6 ρ14 2 2 -2 0 2 2 -2 0 0 0 2 -2 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 -1 -2 -2 0 0 0 -1 -1 0 0 1 1 1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ16 2 2 2 0 -1 -2 -2 0 0 0 -1 -1 0 0 1 1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ17 2 2 -2 0 -1 2 -2 0 0 0 -1 1 0 0 1 1 -1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ18 2 2 -2 0 -1 2 -2 0 0 0 -1 1 0 0 1 1 -1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ19 4 -4 0 0 4 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ20 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 -2√3 2√3 0 0 0 0 0 symplectic faithful, Schur index 2 ρ21 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 2√3 -2√3 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C8.D6
On 48 points
Generators in S48
```(1 40 27 18 7 46 33 24)(2 47 28 13 8 41 34 19)(3 42 29 20 9 48 35 14)(4 37 30 15 10 43 36 21)(5 44 31 22 11 38 25 16)(6 39 32 17 12 45 26 23)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 44 19 38)(14 37 20 43)(15 42 21 48)(16 47 22 41)(17 40 23 46)(18 45 24 39)(25 28 31 34)(26 33 32 27)(29 36 35 30)```

`G:=sub<Sym(48)| (1,40,27,18,7,46,33,24)(2,47,28,13,8,41,34,19)(3,42,29,20,9,48,35,14)(4,37,30,15,10,43,36,21)(5,44,31,22,11,38,25,16)(6,39,32,17,12,45,26,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,44,19,38)(14,37,20,43)(15,42,21,48)(16,47,22,41)(17,40,23,46)(18,45,24,39)(25,28,31,34)(26,33,32,27)(29,36,35,30)>;`

`G:=Group( (1,40,27,18,7,46,33,24)(2,47,28,13,8,41,34,19)(3,42,29,20,9,48,35,14)(4,37,30,15,10,43,36,21)(5,44,31,22,11,38,25,16)(6,39,32,17,12,45,26,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,44,19,38)(14,37,20,43)(15,42,21,48)(16,47,22,41)(17,40,23,46)(18,45,24,39)(25,28,31,34)(26,33,32,27)(29,36,35,30) );`

`G=PermutationGroup([[(1,40,27,18,7,46,33,24),(2,47,28,13,8,41,34,19),(3,42,29,20,9,48,35,14),(4,37,30,15,10,43,36,21),(5,44,31,22,11,38,25,16),(6,39,32,17,12,45,26,23)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,44,19,38),(14,37,20,43),(15,42,21,48),(16,47,22,41),(17,40,23,46),(18,45,24,39),(25,28,31,34),(26,33,32,27),(29,36,35,30)]])`

Matrix representation of C8.D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 72 71 0 0 0 1 0 0 0 0 1 0 0 72
,
 0 1 0 0 0 0 72 1 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 27 0 0 0 46 46 0 27
,
 72 1 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 0 27 0 0 0 0 27 27 46 19 0 0 46 0 0 27

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,1,1,0,0,0,1,72,0,0,0,0,0,71,0,72],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,46,0,0,46,0,0,0,46,0,46,0,0,0,0,27,0,0,0,0,0,0,27],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,46,0,27,46,0,0,0,27,27,0,0,0,0,0,46,0,0,0,0,0,19,27] >;`

C8.D6 in GAP, Magma, Sage, TeX

`C_8.D_6`
`% in TeX`

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

`G:=SmallGroup(96,116);`
`// by ID`

`G=gap.SmallGroup(96,116);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,50,579,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^6=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

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