Aliases: D4.A5, SL2(𝔽5).5C22, C4.A5⋊1C2, C4.1(C2×A5), C22.(C2×A5), C2.4(C22×A5), (C2×SL2(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, C2×C4, D4, D4, Q8, D5, C10, Dic3, C12, D6, C2×C6, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, SL2(𝔽3), Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, 2- 1+4, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×SL2(𝔽3), C4.A4, D4⋊2S3, D4⋊2D5, D4.A4, SL2(𝔽5), C4.A5, C2×SL2(𝔽5), D4.A5
Quotients: C1, C2, C22, A5, C2×A5, C22×A5, 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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 C2×A5 |
| 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