Aliases: D4.A5, SL2(F5).5C22, C4.A5:1C2, C4.1(C2xA5), C22.(C2xA5), C2.4(C22xA5), (C2xSL2(F5)):1C2, SmallGroup(480,957)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(F5) — D4.A5 |
| SL2(F5) — D4.A5 |
Subgroups: 748 in 83 conjugacy classes, 11 normal (7 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2xC4, D4, D4, Q8, D5, C10, Dic3, C12, D6, C2xC6, C2xQ8, C4oD4, Dic5, C20, D10, C2xC10, SL2(F3), Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, 2- 1+4, Dic10, C4xD5, C2xDic5, C5:D4, C5xD4, C2xSL2(F3), C4.A4, D4:2S3, D4:2D5, D4.A4, SL2(F5), C4.A5, C2xSL2(F5), D4.A5
Quotients: C1, C2, C22, A5, C2xA5, C22xA5, D4.A5
Character table of D4.A5
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 20A | 20B | |
| size | 1 | 1 | 2 | 2 | 30 | 20 | 2 | 30 | 30 | 30 | 12 | 12 | 20 | 40 | 40 | 12 | 12 | 24 | 24 | 24 | 24 | 40 | 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 | -3 | 3 | 1 | 0 | -3 | 1 | -1 | -1 | 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 | 0 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2xA5 |
| ρ6 | 3 | 3 | 3 | -3 | 1 | 0 | -3 | -1 | 1 | -1 | 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 | 0 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2xA5 |
| ρ7 | 3 | 3 | 3 | 3 | -1 | 0 | 3 | -1 | -1 | -1 | 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 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ8 | 3 | 3 | -3 | -3 | -1 | 0 | 3 | 1 | 1 | -1 | 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 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from C2xA5 |
| ρ9 | 3 | 3 | 3 | -3 | 1 | 0 | -3 | -1 | 1 | -1 | 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 | 0 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2xA5 |
| ρ10 | 3 | 3 | 3 | 3 | -1 | 0 | 3 | -1 | -1 | -1 | 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 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ11 | 3 | 3 | -3 | -3 | -1 | 0 | 3 | 1 | 1 | -1 | 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 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from C2xA5 |
| ρ12 | 3 | 3 | -3 | 3 | 1 | 0 | -3 | 1 | -1 | -1 | 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 | 0 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2xA5 |
| ρ13 | 4 | 4 | -4 | 4 | 0 | 1 | -4 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2xA5 |
| ρ14 | 4 | 4 | 4 | -4 | 0 | 1 | -4 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from C2xA5 |
| ρ15 | 4 | 4 | 4 | 4 | 0 | 1 | 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 | -4 | -4 | 0 | 1 | 4 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2xA5 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ19 | 5 | 5 | -5 | 5 | -1 | -1 | -5 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | orthogonal lifted from C2xA5 |
| ρ20 | 5 | 5 | 5 | -5 | -1 | -1 | -5 | 1 | -1 | 1 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | orthogonal lifted from C2xA5 |
| ρ21 | 5 | 5 | -5 | -5 | 1 | -1 | 5 | -1 | -1 | 1 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from C2xA5 |
| ρ22 | 5 | 5 | 5 | 5 | 1 | -1 | 5 | 1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from A5 |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 24 46 20 11)(2 10 15 43 23 8 4 21 37 17)(5 7 18 40 14)(6 12)(13 47 27 35 45 19 41 33 29 39)(16 44 30 32 48)(22 38 36 26 42)(25 31)
(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)| (1,24,46,20,11)(2,10,15,43,23,8,4,21,37,17)(5,7,18,40,14)(6,12)(13,47,27,35,45,19,41,33,29,39)(16,44,30,32,48)(22,38,36,26,42)(25,31), (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( (1,24,46,20,11)(2,10,15,43,23,8,4,21,37,17)(5,7,18,40,14)(6,12)(13,47,27,35,45,19,41,33,29,39)(16,44,30,32,48)(22,38,36,26,42)(25,31), (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([[(1,24,46,20,11),(2,10,15,43,23,8,4,21,37,17),(5,7,18,40,14),(6,12),(13,47,27,35,45,19,41,33,29,39),(16,44,30,32,48),(22,38,36,26,42),(25,31)], [(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 D4.A5 ►in GL4(F5) generated by
| 0 | 1 | 0 | 2 |
| 4 | 0 | 2 | 0 |
| 0 | 4 | 0 | 4 |
| 1 | 0 | 2 | 0 |
| 2 | 0 | 4 | 0 |
| 0 | 4 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 2 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,4,0,1,1,0,4,0,0,2,0,2,2,0,4,0],[2,0,4,0,0,4,0,2,4,0,0,0,0,1,0,4] >;
D4.A5 in GAP, Magma, Sage, TeX
D_4.A_5
% in TeX
G:=Group("D4.A5"); // GroupNames label
G:=SmallGroup(480,957);
// by ID
G=gap.SmallGroup(480,957);
# by ID
Export