Aliases: A5⋊C8, C4.4S5, (C2×A5).C4, (C4×A5).3C2, C2.1(A5⋊C4), SmallGroup(480,217)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — A5⋊C8 |
| A5 — A5⋊C8 |
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 |
►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
Subgroup lattice of A5⋊C8 in TeX
Character table of A5⋊C8 in TeX