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## G = C8.A5order 480 = 25·3·5

### The central extension by C8 of A5

Aliases: C8.A5, SL2(𝔽5).C4, C4.5(C2×A5), C2.2(C4×A5), C4.A5.3C2, SmallGroup(480,221)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C4 — C8 — C8.A5
 Derived series SL2(𝔽5) — C8.A5
 Lower central SL2(𝔽5) — C8.A5
 Upper central C1 — C8

30C2
10C3
6C5
15C22
15C4
10S3
10S3
10C6
6D5
6C10
6D5
5Q8
15D4
15C8
15C2×C4
10C12
10Dic3
10D6
6C20
6Dic5
6D10
15C2×C8
15M4(2)
10C4×S3
10C3⋊C8
10C24
6C40
10S3×C8

Smallest permutation representation of C8.A5
On 48 points
Generators in S48
```(1 10 44 3 20 14 5 30 24 7 40 34)(2 35 29 4 45 39 6 15 9 8 25 19)(11 27 23 21 37 33 31 47 43 41 17 13)(12 48 46 22 18 16 32 28 26 42 38 36)
(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)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 5A 5B 6 8A 8B 8C 8D 8E 8F 10A 10B 12A 12B 20A 20B 20C 20D 24A 24B 24C 24D 40A ··· 40H order 1 2 2 3 4 4 4 5 5 6 8 8 8 8 8 8 10 10 12 12 20 20 20 20 24 24 24 24 40 ··· 40 size 1 1 30 20 1 1 30 12 12 20 1 1 1 1 30 30 12 12 20 20 12 12 12 12 20 20 20 20 12 ··· 12

36 irreducible representations

 dim 1 1 1 2 3 3 3 4 4 4 4 5 5 5 6 type + + + + + + + + image C1 C2 C4 C8.A5 A5 C2×A5 C4×A5 A5 C2×A5 C4×A5 C8.A5 A5 C2×A5 C4×A5 C8.A5 kernel C8.A5 C4.A5 SL2(𝔽5) C1 C8 C4 C2 C8 C4 C2 C1 C8 C4 C2 C1 # reps 1 1 2 8 2 2 4 1 1 2 4 1 1 2 4

Matrix representation of C8.A5 in GL2(𝔽41) generated by

 30 32 16 2
,
 22 13 31 3
`G:=sub<GL(2,GF(41))| [30,16,32,2],[22,31,13,3] >;`

C8.A5 in GAP, Magma, Sage, TeX

`C_8.A_5`
`% in TeX`

`G:=Group("C8.A5");`
`// GroupNames label`

`G:=SmallGroup(480,221);`
`// by ID`

`G=gap.SmallGroup(480,221);`
`# by ID`

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