non-abelian, perfect, quasisimple, not soluble
Aliases: SL2(𝔽5), SU2(𝔽5), Spin3(𝔽5), C2.A5, Binary icosahedral group (2I or <2,3,5>), SmallGroup(120,5)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) |
| SL2(𝔽5) |
Character table of SL2(𝔽5)
| class | 1 | 2 | 3 | 4 | 5A | 5B | 6 | 10A | 10B | |
| size | 1 | 1 | 20 | 30 | 12 | 12 | 20 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 2 | -2 | -1 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | symplectic faithful, Schur index 2 |
| ρ3 | 2 | -2 | -1 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | symplectic faithful, Schur index 2 |
| ρ4 | 3 | 3 | 0 | -1 | 1-√5/2 | 1+√5/2 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ5 | 3 | 3 | 0 | -1 | 1+√5/2 | 1-√5/2 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ6 | 4 | 4 | 1 | 0 | -1 | -1 | 1 | -1 | -1 | orthogonal lifted from A5 |
| ρ7 | 4 | -4 | 1 | 0 | -1 | -1 | -1 | 1 | 1 | symplectic faithful, Schur index 2 |
| ρ8 | 5 | 5 | -1 | 1 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from A5 |
| ρ9 | 6 | -6 | 0 | 0 | 1 | 1 | 0 | -1 | -1 | symplectic faithful, Schur index 2 |
►On 24 points - transitive group 24T201
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 14 19)(2 11 18)(3 16 17)(4 9 20)(5 6 12)(7 8 10)(13 24 21)(15 22 23)
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), (1,14,19)(2,11,18)(3,16,17)(4,9,20)(5,6,12)(7,8,10)(13,24,21)(15,22,23)>;
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), (1,14,19)(2,11,18)(3,16,17)(4,9,20)(5,6,12)(7,8,10)(13,24,21)(15,22,23) );
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)], [(1,14,19),(2,11,18),(3,16,17),(4,9,20),(5,6,12),(7,8,10),(13,24,21),(15,22,23)]])
G:=TransitiveGroup(24,201);
SL2(𝔽5) is a maximal subgroup of
CSU2(𝔽5) C2.S5 C4.A5
Matrix representation of SL2(𝔽5) ►in GL2(𝔽5) generated by
| 4 | 2 |
| 4 | 1 |
| 3 | 3 |
| 4 | 1 |
G:=sub<GL(2,GF(5))| [4,4,2,1],[3,4,3,1] >;
SL2(𝔽5) in GAP, Magma, Sage, TeX
{\rm SL}_2({\mathbb F}_5) % in TeX
G:=Group("SL(2,5)"); // GroupNames label
G:=SmallGroup(120,5);
// by ID
G=gap.SmallGroup(120,5);
# by ID
Export
Subgroup lattice of SL2(𝔽5) in TeX
Character table of SL2(𝔽5) in TeX