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## G = C12.C8order 96 = 25·3

### 1st non-split extension by C12 of C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C12.C8
 Chief series C1 — C3 — C6 — C12 — C24 — C3⋊C16 — C12.C8
 Lower central C3 — C6 — C12.C8
 Upper central C1 — C8 — C2×C8

Generators and relations for C12.C8
G = < a,b | a24=1, b4=a18, bab-1=a5 >

Smallest permutation representation of C12.C8
On 48 points
Generators in S48
```(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 41 10 38 19 35 4 32 13 29 22 26 7 47 16 44)(2 46 11 43 20 40 5 37 14 34 23 31 8 28 17 25)(3 27 12 48 21 45 6 42 15 39 24 36 9 33 18 30)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 6A 6B 6C 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 16A ··· 16H 24A ··· 24H order 1 2 2 3 4 4 4 6 6 6 8 8 8 8 8 8 12 12 12 12 16 ··· 16 24 ··· 24 size 1 1 2 2 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 6 ··· 6 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 S3 Dic3 D6 Dic3 C3⋊C8 C3⋊C8 M5(2) C12.C8 kernel C12.C8 C3⋊C16 C2×C24 C24 C2×C12 C12 C2×C6 C2×C8 C8 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 1 2 2 4 4 1 1 1 1 2 2 4 8

Matrix representation of C12.C8 in GL2(𝔽97) generated by

 73 0 0 9
,
 0 1 33 0
`G:=sub<GL(2,GF(97))| [73,0,0,9],[0,33,1,0] >;`

C12.C8 in GAP, Magma, Sage, TeX

`C_{12}.C_8`
`% in TeX`

`G:=Group("C12.C8");`
`// GroupNames label`

`G:=SmallGroup(96,19);`
`// by ID`

`G=gap.SmallGroup(96,19);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,50,69,2309]);`
`// Polycyclic`

`G:=Group<a,b|a^24=1,b^4=a^18,b*a*b^-1=a^5>;`
`// generators/relations`

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