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## G = C24.1C8order 192 = 26·3

### 1st non-split extension by C24 of C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C24.1C8
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C12.C8 — C24.1C8
 Lower central C3 — C6 — C12 — C24.1C8
 Upper central C1 — C8 — C2×C8 — C4×C8

Generators and relations for C24.1C8
G = < a,b | a24=1, b8=a12, bab-1=a-1 >

Smallest permutation representation of C24.1C8
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 33 22 36 19 39 16 42 13 45 10 48 7 27 4 30)(2 32 23 35 20 38 17 41 14 44 11 47 8 26 5 29)(3 31 24 34 21 37 18 40 15 43 12 46 9 25 6 28)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C ··· 4G 6A 6B 6C 8A 8B 8C 8D 8E ··· 8J 12A ··· 12L 16A ··· 16H 24A ··· 24P order 1 2 2 3 4 4 4 ··· 4 6 6 6 8 8 8 8 8 ··· 8 12 ··· 12 16 ··· 16 24 ··· 24 size 1 1 2 2 1 1 2 ··· 2 2 2 2 1 1 1 1 2 ··· 2 2 ··· 2 12 ··· 12 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - - + - + image C1 C2 C2 C4 C4 C8 S3 D4 Q8 Dic3 Dic3 D6 M4(2) C3⋊C8 Dic6 D12 C4.Dic3 C8.C8 C24.1C8 kernel C24.1C8 C12.C8 C4×C24 C4×C12 C2×C24 C24 C4×C8 C24 C24 C42 C2×C8 C2×C8 C2×C6 C8 C8 C8 C22 C3 C1 # reps 1 2 1 2 2 8 1 1 1 1 1 1 2 4 2 2 4 8 16

Matrix representation of C24.1C8 in GL2(𝔽97) generated by

 4 0 0 73
,
 0 1 47 0
`G:=sub<GL(2,GF(97))| [4,0,0,73],[0,47,1,0] >;`

C24.1C8 in GAP, Magma, Sage, TeX

`C_{24}._1C_8`
`% in TeX`

`G:=Group("C24.1C8");`
`// GroupNames label`

`G:=SmallGroup(192,22);`
`// by ID`

`G=gap.SmallGroup(192,22);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,64,100,1123,136,102,6278]);`
`// Polycyclic`

`G:=Group<a,b|a^24=1,b^8=a^12,b*a*b^-1=a^-1>;`
`// generators/relations`

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