Copied to
clipboard

## G = C48.C4order 192 = 26·3

### 1st non-split extension by C48 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C48.C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C24.C4 — C48.C4
 Lower central C3 — C6 — C12 — C24 — C48.C4
 Upper central C1 — C4 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C48.C4
G = < a,b | a48=1, b4=a24, bab-1=a23 >

Smallest permutation representation of C48.C4
On 96 points
Generators in S96
```(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 55 13 91 25 79 37 67)(2 78 14 66 26 54 38 90)(3 53 15 89 27 77 39 65)(4 76 16 64 28 52 40 88)(5 51 17 87 29 75 41 63)(6 74 18 62 30 50 42 86)(7 49 19 85 31 73 43 61)(8 72 20 60 32 96 44 84)(9 95 21 83 33 71 45 59)(10 70 22 58 34 94 46 82)(11 93 23 81 35 69 47 57)(12 68 24 56 36 92 48 80)```

`G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,55,13,91,25,79,37,67)(2,78,14,66,26,54,38,90)(3,53,15,89,27,77,39,65)(4,76,16,64,28,52,40,88)(5,51,17,87,29,75,41,63)(6,74,18,62,30,50,42,86)(7,49,19,85,31,73,43,61)(8,72,20,60,32,96,44,84)(9,95,21,83,33,71,45,59)(10,70,22,58,34,94,46,82)(11,93,23,81,35,69,47,57)(12,68,24,56,36,92,48,80)>;`

`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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,55,13,91,25,79,37,67)(2,78,14,66,26,54,38,90)(3,53,15,89,27,77,39,65)(4,76,16,64,28,52,40,88)(5,51,17,87,29,75,41,63)(6,74,18,62,30,50,42,86)(7,49,19,85,31,73,43,61)(8,72,20,60,32,96,44,84)(9,95,21,83,33,71,45,59)(10,70,22,58,34,94,46,82)(11,93,23,81,35,69,47,57)(12,68,24,56,36,92,48,80) );`

`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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,55,13,91,25,79,37,67),(2,78,14,66,26,54,38,90),(3,53,15,89,27,77,39,65),(4,76,16,64,28,52,40,88),(5,51,17,87,29,75,41,63),(6,74,18,62,30,50,42,86),(7,49,19,85,31,73,43,61),(8,72,20,60,32,96,44,84),(9,95,21,83,33,71,45,59),(10,70,22,58,34,94,46,82),(11,93,23,81,35,69,47,57),(12,68,24,56,36,92,48,80)]])`

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + + - - + + - image C1 C2 C2 C4 S3 Q8 D4 Dic3 D6 D8 Q16 Dic6 D12 D24 Dic12 C8.4Q8 C48.C4 kernel C48.C4 C24.C4 C2×C48 C48 C2×C16 C24 C2×C12 C16 C2×C8 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 1 4 1 1 1 2 1 2 2 2 2 4 4 8 16

Matrix representation of C48.C4 in GL2(𝔽97) generated by

 11 0 0 44
,
 0 1 75 0
`G:=sub<GL(2,GF(97))| [11,0,0,44],[0,75,1,0] >;`

C48.C4 in GAP, Magma, Sage, TeX

`C_{48}.C_4`
`% in TeX`

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

`G:=SmallGroup(192,65);`
`// by ID`

`G=gap.SmallGroup(192,65);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,176,184,675,192,1684,102,6278]);`
`// Polycyclic`

`G:=Group<a,b|a^48=1,b^4=a^24,b*a*b^-1=a^23>;`
`// generators/relations`

Export

׿
×
𝔽