Aliases: GL2(𝔽5), C4.5S5, SL2(𝔽5)⋊1C4, C4.A5.2C2, C2.2(A5⋊C4), Aut(C52), SmallGroup(480,218)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — GL2(𝔽5) |
| SL2(𝔽5) — GL2(𝔽5) |
30C2
10C3
6C5
15C22
15C4
30C4
30C4
10C6
20S3
6C10
6D5
6D5
5Q8
10C8
15D4
15C2×C4
30C2×C4
10C12
10D6
10Dic3
6F5
6F5
6C20
6F5
6F5
6D10
6Dic5
15C42
15M4(2)
10C3⋊C8
10C4×S3
10C24
15C4≀C2
10C8⋊S3
Character table of GL2(𝔽5)
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5 | 6 | 8A | 8B | 10 | 12A | 12B | 20A | 20B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 30 | 20 | 1 | 1 | 30 | 30 | 30 | 30 | 30 | 24 | 20 | 20 | 20 | 24 | 20 | 20 | 24 | 24 | 20 | 20 | 20 | 20 | |
| 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 | i | -i | i | -i | 1 | 1 | 1 | i | -i | 1 | -1 | -1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | -i | i | 1 | -1 | -1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ5 | 4 | 4 | 0 | 1 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 2 | 2 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ6 | 4 | 4 | 0 | 1 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -2 | -2 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from S5 |
| ρ7 | 4 | 4 | 0 | 1 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 2i | -2i | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | complex lifted from A5⋊C4 |
| ρ8 | 4 | 4 | 0 | 1 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -2i | 2i | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | complex lifted from A5⋊C4 |
| ρ9 | 4 | -4 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 1 | 2i | -2i | -i | i | 0 | 0 | 0 | 0 | complex faithful |
| ρ10 | 4 | -4 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 1 | -2i | 2i | i | -i | 0 | 0 | 0 | 0 | complex faithful |
| ρ11 | 4 | -4 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | -i | i | -i | i | 2ζ87ζ3+ζ87 | 2ζ8ζ3+ζ8 | 2ζ83ζ3+ζ83 | 2ζ85ζ3+ζ85 | complex faithful |
| ρ12 | 4 | -4 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | i | -i | i | -i | 2ζ85ζ3+ζ85 | 2ζ83ζ3+ζ83 | 2ζ8ζ3+ζ8 | 2ζ87ζ3+ζ87 | complex faithful |
| ρ13 | 4 | -4 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | -i | i | -i | i | 2ζ83ζ3+ζ83 | 2ζ85ζ3+ζ85 | 2ζ87ζ3+ζ87 | 2ζ8ζ3+ζ8 | complex faithful |
| ρ14 | 4 | -4 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | i | -i | i | -i | 2ζ8ζ3+ζ8 | 2ζ87ζ3+ζ87 | 2ζ85ζ3+ζ85 | 2ζ83ζ3+ζ83 | complex faithful |
| ρ15 | 5 | 5 | 1 | -1 | 5 | 5 | 1 | 1 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | 0 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ16 | 5 | 5 | 1 | -1 | 5 | 5 | -1 | -1 | -1 | -1 | 1 | 0 | -1 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S5 |
| ρ17 | 5 | 5 | -1 | -1 | -5 | -5 | i | -i | i | -i | 1 | 0 | -1 | -i | i | 0 | 1 | 1 | 0 | 0 | i | -i | i | -i | complex lifted from A5⋊C4 |
| ρ18 | 5 | 5 | -1 | -1 | -5 | -5 | -i | i | -i | i | 1 | 0 | -1 | i | -i | 0 | 1 | 1 | 0 | 0 | -i | i | -i | i | complex lifted from A5⋊C4 |
| ρ19 | 6 | 6 | -2 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S5 |
| ρ20 | 6 | 6 | 2 | 0 | -6 | -6 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A5⋊C4 |
| ρ21 | 6 | -6 | 0 | 0 | -6i | 6i | -1-i | -1+i | 1+i | 1-i | 0 | 1 | 0 | 0 | 0 | -1 | 0 | 0 | -i | i | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 6 | -6 | 0 | 0 | 6i | -6i | -1+i | -1-i | 1-i | 1+i | 0 | 1 | 0 | 0 | 0 | -1 | 0 | 0 | i | -i | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 6 | -6 | 0 | 0 | 6i | -6i | 1-i | 1+i | -1+i | -1-i | 0 | 1 | 0 | 0 | 0 | -1 | 0 | 0 | i | -i | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 6 | -6 | 0 | 0 | -6i | 6i | 1+i | 1-i | -1-i | -1+i | 0 | 1 | 0 | 0 | 0 | -1 | 0 | 0 | -i | i | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T1353
Generators in S24
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)
(1 7 13 19)(3 24 10 23)(4 17 21 18)(5 9 6 16)(11 15 12 22)
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,7,13,19)(3,24,10,23)(4,17,21,18)(5,9,6,16)(11,15,12,22)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,7,13,19)(3,24,10,23)(4,17,21,18)(5,9,6,16)(11,15,12,22) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)], [(1,7,13,19),(3,24,10,23),(4,17,21,18),(5,9,6,16),(11,15,12,22)]])
G:=TransitiveGroup(24,1353);
Matrix representation of GL2(𝔽5) ►in GL2(𝔽5) generated by
| 1 | 4 |
| 2 | 1 |
| 0 | 3 |
| 4 | 1 |
G:=sub<GL(2,GF(5))| [1,2,4,1],[0,4,3,1] >;
GL2(𝔽5) in GAP, Magma, Sage, TeX
{\rm GL}_2({\mathbb F}_5) % in TeX
G:=Group("GL(2,5)"); // GroupNames label
G:=SmallGroup(480,218);
// by ID
G=gap.SmallGroup(480,218);
# by ID
Export
Subgroup lattice of GL2(𝔽5) in TeX
Character table of GL2(𝔽5) in TeX