Copied to
clipboard

## G = A5⋊C8order 480 = 25·3·5

### The semidirect product of A5 and C8 acting via C8/C4=C2

Aliases: A5⋊C8, C4.4S5, (C2×A5).C4, (C4×A5).3C2, C2.1(A5⋊C4), SmallGroup(480,217)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C4 — C4×A5 — A5⋊C8
 Derived series A5 — A5⋊C8
 Lower central A5 — A5⋊C8
 Upper central C1 — C4

15C2
15C2
10C3
6C5
5C22
15C22
15C4
15C22
10C6
10S3
10S3
6D5
6D5
6C10
5C23
10C8
15C2×C4
15C2×C4
30C8
5A4
10C12
10Dic3
10D6
6Dic5
6C20
6D10
15C2×C8
15C2×C8
10C3⋊C8
10C4×S3
10C24
15C22⋊C8
10S3×C8

Character table of A5⋊C8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6 8A 8B 8C 8D 8E 8F 8G 8H 10 12A 12B 20A 20B 24A 24B 24C 24D size 1 1 15 15 20 1 1 15 15 24 20 10 10 10 10 30 30 30 30 24 20 20 24 24 20 20 20 20 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 -1 -1 1 1 -i i i -i i -i i -i 1 -1 -1 -1 -1 i i -i -i linear of order 4 ρ4 1 1 1 1 1 -1 -1 -1 -1 1 1 i -i -i i -i i -i i 1 -1 -1 -1 -1 -i -i i i linear of order 4 ρ5 1 -1 -1 1 1 i -i i -i 1 -1 ζ83 ζ8 ζ85 ζ87 ζ85 ζ87 ζ8 ζ83 -1 -i i -i i ζ8 ζ85 ζ87 ζ83 linear of order 8 ρ6 1 -1 -1 1 1 i -i i -i 1 -1 ζ87 ζ85 ζ8 ζ83 ζ8 ζ83 ζ85 ζ87 -1 -i i -i i ζ85 ζ8 ζ83 ζ87 linear of order 8 ρ7 1 -1 -1 1 1 -i i -i i 1 -1 ζ85 ζ87 ζ83 ζ8 ζ83 ζ8 ζ87 ζ85 -1 i -i i -i ζ87 ζ83 ζ8 ζ85 linear of order 8 ρ8 1 -1 -1 1 1 -i i -i i 1 -1 ζ8 ζ83 ζ87 ζ85 ζ87 ζ85 ζ83 ζ8 -1 i -i i -i ζ83 ζ87 ζ85 ζ8 linear of order 8 ρ9 4 4 0 0 1 4 4 0 0 -1 1 2 2 2 2 0 0 0 0 -1 1 1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S5 ρ10 4 4 0 0 1 4 4 0 0 -1 1 -2 -2 -2 -2 0 0 0 0 -1 1 1 -1 -1 1 1 1 1 orthogonal lifted from S5 ρ11 4 4 0 0 1 -4 -4 0 0 -1 1 2i -2i -2i 2i 0 0 0 0 -1 -1 -1 1 1 i i -i -i complex lifted from A5⋊C4 ρ12 4 4 0 0 1 -4 -4 0 0 -1 1 -2i 2i 2i -2i 0 0 0 0 -1 -1 -1 1 1 -i -i i i complex lifted from A5⋊C4 ρ13 4 -4 0 0 1 -4i 4i 0 0 -1 -1 2ζ8 2ζ83 2ζ87 2ζ85 0 0 0 0 1 i -i -i i ζ87 ζ83 ζ8 ζ85 complex faithful ρ14 4 -4 0 0 1 4i -4i 0 0 -1 -1 2ζ83 2ζ8 2ζ85 2ζ87 0 0 0 0 1 -i i i -i ζ85 ζ8 ζ83 ζ87 complex faithful ρ15 4 -4 0 0 1 4i -4i 0 0 -1 -1 2ζ87 2ζ85 2ζ8 2ζ83 0 0 0 0 1 -i i i -i ζ8 ζ85 ζ87 ζ83 complex faithful ρ16 4 -4 0 0 1 -4i 4i 0 0 -1 -1 2ζ85 2ζ87 2ζ83 2ζ8 0 0 0 0 1 i -i -i i ζ83 ζ87 ζ85 ζ8 complex faithful ρ17 5 5 1 1 -1 5 5 1 1 0 -1 -1 -1 -1 -1 1 1 1 1 0 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S5 ρ18 5 5 1 1 -1 5 5 1 1 0 -1 1 1 1 1 -1 -1 -1 -1 0 -1 -1 0 0 1 1 1 1 orthogonal lifted from S5 ρ19 5 5 1 1 -1 -5 -5 -1 -1 0 -1 -i i i -i -i i -i i 0 1 1 0 0 i i -i -i complex lifted from A5⋊C4 ρ20 5 5 1 1 -1 -5 -5 -1 -1 0 -1 i -i -i i i -i i -i 0 1 1 0 0 -i -i i i complex lifted from A5⋊C4 ρ21 5 -5 -1 1 -1 -5i 5i -i i 0 1 ζ8 ζ83 ζ87 ζ85 ζ83 ζ8 ζ87 ζ85 0 -i i 0 0 ζ83 ζ87 ζ85 ζ8 complex faithful ρ22 5 -5 -1 1 -1 -5i 5i -i i 0 1 ζ85 ζ87 ζ83 ζ8 ζ87 ζ85 ζ83 ζ8 0 -i i 0 0 ζ87 ζ83 ζ8 ζ85 complex faithful ρ23 5 -5 -1 1 -1 5i -5i i -i 0 1 ζ87 ζ85 ζ8 ζ83 ζ85 ζ87 ζ8 ζ83 0 i -i 0 0 ζ85 ζ8 ζ83 ζ87 complex faithful ρ24 5 -5 -1 1 -1 5i -5i i -i 0 1 ζ83 ζ8 ζ85 ζ87 ζ8 ζ83 ζ85 ζ87 0 i -i 0 0 ζ8 ζ85 ζ87 ζ83 complex faithful ρ25 6 6 -2 -2 0 6 6 -2 -2 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 orthogonal lifted from S5 ρ26 6 6 -2 -2 0 -6 -6 2 2 1 0 0 0 0 0 0 0 0 0 1 0 0 -1 -1 0 0 0 0 orthogonal lifted from A5⋊C4 ρ27 6 -6 2 -2 0 6i -6i -2i 2i 1 0 0 0 0 0 0 0 0 0 -1 0 0 -i i 0 0 0 0 complex faithful, Schur index 2 ρ28 6 -6 2 -2 0 -6i 6i 2i -2i 1 0 0 0 0 0 0 0 0 0 -1 0 0 i -i 0 0 0 0 complex faithful, Schur index 2

Smallest permutation representation of A5⋊C8
On 40 points
Generators in S40
```(1 10 23 40 3 12 17 34)(2 11 20 37 4 9 26 31)(5 38 28 15 7 32 22 13)(6 35 25 16 8 29 19 14)(18 39 21 30 24 33 27 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)```

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

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

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

Matrix representation of A5⋊C8 in GL5(𝔽241)

 233 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 177 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1

`G:=sub<GL(5,GF(241))| [233,0,0,0,0,0,0,1,0,240,0,0,0,0,240,0,0,0,1,240,0,1,0,0,240],[177,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

A5⋊C8 in GAP, Magma, Sage, TeX

`A_5\rtimes C_8`
`% in TeX`

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

`G:=SmallGroup(480,217);`
`// by ID`

`G=gap.SmallGroup(480,217);`
`# by ID`

Export

׿
×
𝔽