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## G = D6⋊C8order 96 = 25·3

### The semidirect product of D6 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D6⋊C8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — D6⋊C8
 Lower central C3 — C6 — D6⋊C8
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for D6⋊C8
G = < a,b,c | a6=b2=c8=1, bab=a-1, ac=ca, cbc-1=a3b >

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 24 ··· 24 size 1 1 1 1 6 6 2 1 1 1 1 6 6 2 2 2 2 2 2 2 6 6 6 6 2 2 2 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D4 D6 M4(2) D12 C3⋊D4 C4×S3 S3×C8 C8⋊S3 kernel D6⋊C8 C2×C3⋊C8 C2×C24 S3×C2×C4 C2×Dic3 C22×S3 D6 C2×C8 C12 C2×C4 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 2 8 1 2 1 2 2 2 2 4 4

Matrix representation of D6⋊C8 in GL3(𝔽73) generated by

 1 0 0 0 1 1 0 72 0
,
 1 0 0 0 72 72 0 0 1
,
 22 0 0 0 8 16 0 57 65
`G:=sub<GL(3,GF(73))| [1,0,0,0,1,72,0,1,0],[1,0,0,0,72,0,0,72,1],[22,0,0,0,8,57,0,16,65] >;`

D6⋊C8 in GAP, Magma, Sage, TeX

`D_6\rtimes C_8`
`% in TeX`

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

`G:=SmallGroup(96,27);`
`// by ID`

`G=gap.SmallGroup(96,27);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,86,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;`
`// generators/relations`

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