Aliases: Q8.A5, SL2(𝔽5)⋊4C22, C4.A5⋊2C2, C4.3(C2×A5), C2.6(C22×A5), 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, C2×C4, D4, Q8, Q8, C23, D5, C10, Dic3, C12, D6, C2×D4, C4○D4, Dic5, C20, D10, SL2(𝔽3), C4×S3, D12, C3×Q8, 2+ 1+4, C4×D5, D20, C5×Q8, C4.A4, Q8⋊3S3, Q8⋊2D5, Q8.A4, SL2(𝔽5), C4.A5, Q8.A5
Quotients: C1, C2, C22, A5, C2×A5, C22×A5, 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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