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## G = C48⋊C4order 192 = 26·3

### 2nd semidirect product of C48 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C48⋊C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C24⋊C4 — C48⋊C4
 Lower central C3 — C12 — C48⋊C4
 Upper central C1 — C4 — M5(2)

Generators and relations for C48⋊C4
G = < a,b | a48=b4=1, bab-1=a29 >

Smallest permutation representation of C48⋊C4
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)
(2 6 26 30)(3 11)(4 16 28 40)(5 21)(7 31)(8 36 32 12)(9 41)(10 46 34 22)(14 18 38 42)(15 23)(17 33)(19 43)(20 48 44 24)(27 35)(29 45)(39 47)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 8D 8E 8F 12A 12B 12C 16A 16B 16C 16D 16E 16F 16G 16H 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 3 4 4 4 4 4 6 6 8 8 8 8 8 8 12 12 12 16 16 16 16 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 2 1 1 2 12 12 2 4 2 2 2 2 12 12 2 2 4 4 4 4 4 12 12 12 12 2 2 2 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + image C1 C2 C2 C2 C4 C4 C4 C4 S3 Dic3 D6 M4(2) M4(2) C4×S3 C4×S3 C8⋊S3 C8⋊S3 C16⋊C4 C48⋊C4 kernel C48⋊C4 C12.C8 C24⋊C4 C3×M5(2) C3⋊C16 C48 C2×C3⋊C8 C4×Dic3 M5(2) C16 C2×C8 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 4 4 2 2 1 2 1 2 2 2 2 4 4 2 4

Matrix representation of C48⋊C4 in GL4(𝔽5) generated by

 0 1 1 0 2 0 0 0 4 0 0 3 0 4 2 0
,
 4 0 0 0 0 3 4 0 0 0 2 0 0 0 0 1
`G:=sub<GL(4,GF(5))| [0,2,4,0,1,0,0,4,1,0,0,2,0,0,3,0],[4,0,0,0,0,3,0,0,0,4,2,0,0,0,0,1] >;`

C48⋊C4 in GAP, Magma, Sage, TeX

`C_{48}\rtimes C_4`
`% in TeX`

`G:=Group("C48:C4");`
`// GroupNames label`

`G:=SmallGroup(192,71);`
`// by ID`

`G=gap.SmallGroup(192,71);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1123,80,102,6278]);`
`// Polycyclic`

`G:=Group<a,b|a^48=b^4=1,b*a*b^-1=a^29>;`
`// generators/relations`

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