direct product, non-abelian, not soluble
Aliases: Q8×A5, C4.2(C2×A5), (C4×A5).1C2, C2.5(C22×A5), (C2×A5).8C22, SmallGroup(480,958)
Series: Chief►Derived ►Lower central ►Upper central
Subgroups: 706 in 84 conjugacy classes, 12 normal (6 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, Q8, Q8, C23, D5, C10, Dic3, C12, A4, D6, C22×C4, C2×Q8, Dic5, C20, D10, Dic6, C4×S3, C3×Q8, C2×A4, C22×Q8, Dic10, C4×D5, C5×Q8, C4×A4, S3×Q8, A5, Q8×D5, Q8×A4, C2×A5, C4×A5, Q8×A5
Quotients: C1, C2, C22, Q8, A5, C2×A5, C22×A5, Q8×A5
Character table of Q8×A5
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 12C | 20A | 20B | 20C | 20D | 20E | 20F | |
| size | 1 | 1 | 15 | 15 | 20 | 2 | 2 | 2 | 30 | 30 | 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 | 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 | 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 | -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 | 1 | linear of order 2 |
| ρ5 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ6 | 3 | 3 | -1 | -1 | 0 | -3 | -3 | 3 | -1 | 1 | 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 |
| ρ7 | 3 | 3 | -1 | -1 | 0 | -3 | 3 | -3 | 1 | -1 | 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 | 0 | 3 | -3 | -3 | 1 | 1 | -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 | 0 | 3 | 3 | 3 | -1 | -1 | -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 |
| ρ10 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | 3 | -1 | -1 | -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 |
| ρ11 | 3 | 3 | -1 | -1 | 0 | 3 | -3 | -3 | 1 | 1 | -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 | 0 | -3 | -3 | 3 | -1 | 1 | 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 |
| ρ13 | 3 | 3 | -1 | -1 | 0 | -3 | 3 | -3 | 1 | -1 | 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 |
| ρ14 | 4 | 4 | 0 | 0 | 1 | 4 | -4 | -4 | 0 | 0 | 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 | 1 | 4 | 4 | 4 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ16 | 4 | 4 | 0 | 0 | 1 | -4 | 4 | -4 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×A5 |
| ρ17 | 4 | 4 | 0 | 0 | 1 | -4 | -4 | 4 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×A5 |
| ρ18 | 5 | 5 | 1 | 1 | -1 | 5 | 5 | 5 | 1 | 1 | 1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ19 | 5 | 5 | 1 | 1 | -1 | -5 | 5 | -5 | -1 | 1 | -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 | -5 | -5 | 5 | 1 | -1 | -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 | 5 | -5 | -5 | -1 | -1 | 1 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ22 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 0 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ23 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 0 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 8 | -8 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ25 | 10 | -10 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 40 points
Generators in S40
(1 32 19 38 5 36 11 22 9 28 15 26)(2 39 12 29)(3 30 13 40)(4 33 10 31 16 37 14 23 20 21 6 27)(7 34 17 24)(8 25 18 35)
(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)| (1,32,19,38,5,36,11,22,9,28,15,26)(2,39,12,29)(3,30,13,40)(4,33,10,31,16,37,14,23,20,21,6,27)(7,34,17,24)(8,25,18,35), (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( (1,32,19,38,5,36,11,22,9,28,15,26)(2,39,12,29)(3,30,13,40)(4,33,10,31,16,37,14,23,20,21,6,27)(7,34,17,24)(8,25,18,35), (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([[(1,32,19,38,5,36,11,22,9,28,15,26),(2,39,12,29),(3,30,13,40),(4,33,10,31,16,37,14,23,20,21,6,27),(7,34,17,24),(8,25,18,35)], [(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 Q8×A5 ►in GL5(𝔽61)
| 2 | 42 | 0 | 0 | 0 |
| 42 | 59 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 60 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 52 |
| 0 | 0 | 9 | 0 | 13 |
| 0 | 0 | 31 | 60 | 22 |
G:=sub<GL(5,GF(61))| [2,42,0,0,0,42,59,0,0,0,0,0,0,0,60,0,0,0,60,0,0,0,1,1,1],[0,60,0,0,0,1,0,0,0,0,0,0,22,9,31,0,0,0,0,60,0,0,52,13,22] >;
Q8×A5 in GAP, Magma, Sage, TeX
Q_8\times A_5
% in TeX
G:=Group("Q8xA5"); // GroupNames label
G:=SmallGroup(480,958);
// by ID
G=gap.SmallGroup(480,958);
# by ID
Export