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## G = D6.C8order 96 = 25·3

### The non-split extension by D6 of C8 acting via C8/C4=C2

Aliases: D6.C8, C486C2, C163S3, C8.20D6, Dic3.C8, C31M5(2), C24.24C22, C3⋊C164C2, C3⋊C8.2C4, C2.3(S3×C8), C6.2(C2×C8), (S3×C8).2C2, (C4×S3).2C4, C4.17(C4×S3), C12.22(C2×C4), SmallGroup(96,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D6.C8
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — D6.C8
 Lower central C3 — C6 — D6.C8
 Upper central C1 — C8 — C16

Generators and relations for D6.C8
G = < a,b,c | a6=b2=1, c8=a3, bab=a-1, ac=ca, cbc-1=a3b >

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

D6.C8 is a maximal subgroup of
D12.4C8  S3×M5(2)  C16.12D6  D8⋊D6  D48⋊C2  SD32⋊S3  Q32⋊S3  C16⋊D9  C24.61D6  C24.62D6  C48⋊S3  C40.52D6  D30.5C8  C80⋊S3  C15⋊M5(2)  D30.C8
D6.C8 is a maximal quotient of
Dic3⋊C16  C4810C4  D6⋊C16  C16⋊D9  C24.61D6  C24.62D6  C48⋊S3  C40.52D6  D30.5C8  C80⋊S3  C15⋊M5(2)  D30.C8

36 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 6 8A 8B 8C 8D 8E 8F 12A 12B 16A 16B 16C 16D 16E 16F 16G 16H 24A 24B 24C 24D 48A ··· 48H order 1 2 2 3 4 4 4 6 8 8 8 8 8 8 12 12 16 16 16 16 16 16 16 16 24 24 24 24 48 ··· 48 size 1 1 6 2 1 1 6 2 1 1 1 1 6 6 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 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 S3 D6 C4×S3 M5(2) S3×C8 D6.C8 kernel D6.C8 C3⋊C16 C48 S3×C8 C3⋊C8 C4×S3 Dic3 D6 C16 C8 C4 C3 C2 C1 # reps 1 1 1 1 2 2 4 4 1 1 2 4 4 8

Matrix representation of D6.C8 in GL2(𝔽41) generated by

 40 38 1 2
,
 40 0 1 1
,
 3 6 39 38
`G:=sub<GL(2,GF(41))| [40,1,38,2],[40,1,0,1],[3,39,6,38] >;`

D6.C8 in GAP, Magma, Sage, TeX

`D_6.C_8`
`% in TeX`

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

`G:=SmallGroup(96,5);`
`// by ID`

`G=gap.SmallGroup(96,5);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,31,50,69,2309]);`
`// Polycyclic`

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

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