Aliases: Q8.A5, SL2(F5):4C22, C4.A5:2C2, C4.3(C2xA5), C2.6(C22xA5), SmallGroup(480,959)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(F5) — Q8.A5 |
| SL2(F5) — Q8.A5 |
Subgroups: 924 in 85 conjugacy classes, 11 normal (5 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2xC4, D4, Q8, Q8, C23, D5, C10, Dic3, C12, D6, C2xD4, C4oD4, Dic5, C20, D10, SL2(F3), C4xS3, D12, C3xQ8, 2+ 1+4, C4xD5, D20, C5xQ8, C4.A4, Q8:3S3, Q8:2D5, Q8.A4, SL2(F5), C4.A5, Q8.A5
Quotients: C1, C2, C22, A5, C2xA5, C22xA5, Q8.A5
Character table of Q8.A5
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 12C | 20A | 20B | 20C | 20D | 20E | 20F | |
| size | 1 | 1 | 30 | 30 | 30 | 20 | 2 | 2 | 2 | 30 | 12 | 12 | 20 | 12 | 12 | 40 | 40 | 40 | 24 | 24 | 24 | 24 | 24 | 24 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 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 |
| ρ4 | 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 |
| ρ5 | 3 | 3 | 1 | 1 | -1 | 0 | 3 | -3 | -3 | -1 | 1+√5/2 | 1-√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from C2xA5 |
| ρ6 | 3 | 3 | -1 | -1 | -1 | 0 | 3 | 3 | 3 | -1 | 1+√5/2 | 1-√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ7 | 3 | 3 | -1 | 1 | 1 | 0 | -3 | -3 | 3 | -1 | 1-√5/2 | 1+√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2xA5 |
| ρ8 | 3 | 3 | 1 | -1 | 1 | 0 | -3 | 3 | -3 | -1 | 1-√5/2 | 1+√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2xA5 |
| ρ9 | 3 | 3 | 1 | 1 | -1 | 0 | 3 | -3 | -3 | -1 | 1-√5/2 | 1+√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from C2xA5 |
| ρ10 | 3 | 3 | -1 | 1 | 1 | 0 | -3 | -3 | 3 | -1 | 1+√5/2 | 1-√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2xA5 |
| ρ11 | 3 | 3 | 1 | -1 | 1 | 0 | -3 | 3 | -3 | -1 | 1+√5/2 | 1-√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2xA5 |
| ρ12 | 3 | 3 | -1 | -1 | -1 | 0 | 3 | 3 | 3 | -1 | 1-√5/2 | 1+√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ13 | 4 | 4 | 0 | 0 | 0 | 1 | 4 | -4 | -4 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2xA5 |
| ρ14 | 4 | 4 | 0 | 0 | 0 | 1 | -4 | -4 | 4 | 0 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2xA5 |
| ρ15 | 4 | 4 | 0 | 0 | 0 | 1 | -4 | 4 | -4 | 0 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2xA5 |
| ρ16 | 4 | 4 | 0 | 0 | 0 | 1 | 4 | 4 | 4 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ19 | 5 | 5 | -1 | 1 | -1 | -1 | -5 | 5 | -5 | 1 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA5 |
| ρ20 | 5 | 5 | -1 | -1 | 1 | -1 | 5 | -5 | -5 | 1 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA5 |
| ρ21 | 5 | 5 | 1 | 1 | 1 | -1 | 5 | 5 | 5 | 1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ22 | 5 | 5 | 1 | -1 | -1 | -1 | -5 | -5 | 5 | 1 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA5 |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
| ρ24 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 48 points
Generators in S48
(1 9 48 6 32 15 3 19 38 8 42 25)(2 24 33 5 37 10 4 14 43 7 47 20)(11 36 30 34 23 17 21 46 40 44 13 27)(12 26 41 45 39 18 22 16 31 35 29 28)
(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)| (1,9,48,6,32,15,3,19,38,8,42,25)(2,24,33,5,37,10,4,14,43,7,47,20)(11,36,30,34,23,17,21,46,40,44,13,27)(12,26,41,45,39,18,22,16,31,35,29,28), (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( (1,9,48,6,32,15,3,19,38,8,42,25)(2,24,33,5,37,10,4,14,43,7,47,20)(11,36,30,34,23,17,21,46,40,44,13,27)(12,26,41,45,39,18,22,16,31,35,29,28), (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([[(1,9,48,6,32,15,3,19,38,8,42,25),(2,24,33,5,37,10,4,14,43,7,47,20),(11,36,30,34,23,17,21,46,40,44,13,27),(12,26,41,45,39,18,22,16,31,35,29,28)], [(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)]])
Matrix representation of Q8.A5 ►in GL4(F5) generated by
| 3 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 3 | 4 | 0 | 0 |
| 2 | 3 | 0 | 0 |
G:=sub<GL(4,GF(5))| [3,2,0,0,3,0,0,0,0,0,3,4,0,0,2,4],[0,0,3,2,0,0,4,3,0,4,0,0,1,0,0,0] >;
Q8.A5 in GAP, Magma, Sage, TeX
Q_8.A_5
% in TeX
G:=Group("Q8.A5"); // GroupNames label
G:=SmallGroup(480,959);
// by ID
G=gap.SmallGroup(480,959);
# by ID
Export