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## G = C12.15C42order 192 = 26·3

### 8th non-split extension by C12 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.15C42
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C12.C8 — C12.15C42
 Lower central C3 — C12 — C12.15C42
 Upper central C1 — C4 — C8⋊C4

Generators and relations for C12.15C42
G = < a,b,c | a12=c4=1, b4=a9, bab-1=a5, ac=ca, cbc-1=a9b >

Smallest permutation representation of C12.15C42
On 48 points
Generators in S48
```(1 29 36 13 25 48 9 21 44 5 17 40)(2 33 18 14 45 30 10 41 26 6 37 22)(3 31 38 15 27 34 11 23 46 7 19 42)(4 35 20 16 47 32 12 43 28 8 39 24)
(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)
(2 14 10 6)(3 11)(4 8 12 16)(7 15)(18 30 26 22)(19 27)(20 24 28 32)(23 31)(33 45 41 37)(34 42)(35 39 43 47)(38 46)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + - - + image C1 C2 C2 C4 C4 C4 S3 Dic3 Dic3 D6 M4(2) M4(2) C4×S3 C4.Dic3 C4.Dic3 C16⋊C4 C12.15C42 kernel C12.15C42 C12.C8 C3×C8⋊C4 C3⋊C16 C4×C12 C2×C24 C8⋊C4 C42 C2×C8 C2×C8 C12 C2×C6 C8 C4 C22 C3 C1 # reps 1 2 1 8 2 2 1 1 1 1 2 2 4 4 4 2 4

Matrix representation of C12.15C42 in GL4(𝔽97) generated by

 91 0 36 46 0 91 0 91 0 0 81 0 0 0 0 81
,
 87 90 40 95 90 50 33 63 59 23 96 83 53 51 55 58
,
 1 55 2 35 0 96 0 65 0 0 22 59 0 0 0 75
`G:=sub<GL(4,GF(97))| [91,0,0,0,0,91,0,0,36,0,81,0,46,91,0,81],[87,90,59,53,90,50,23,51,40,33,96,55,95,63,83,58],[1,0,0,0,55,96,0,0,2,0,22,0,35,65,59,75] >;`

C12.15C42 in GAP, Magma, Sage, TeX

`C_{12}._{15}C_4^2`
`% in TeX`

`G:=Group("C12.15C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,25);`
`// by ID`

`G=gap.SmallGroup(192,25);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,100,1123,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=c^4=1,b^4=a^9,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^9*b>;`
`// generators/relations`

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