non-abelian, perfect, quasisimple, not soluble
Aliases: SL2(𝔽7), SU2(𝔽7), Spin3(𝔽7), C2.GL3(𝔽2), SmallGroup(336,114)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽7) |
| SL2(𝔽7) |
Character table of SL2(𝔽7)
| class | 1 | 2 | 3 | 4 | 6 | 7A | 7B | 8A | 8B | 14A | 14B | |
| size | 1 | 1 | 56 | 42 | 56 | 24 | 24 | 42 | 42 | 24 | 24 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 3 | 3 | 0 | -1 | 0 | -1-√-7/2 | -1+√-7/2 | 1 | 1 | -1-√-7/2 | -1+√-7/2 | complex lifted from GL3(𝔽2) |
| ρ3 | 3 | 3 | 0 | -1 | 0 | -1+√-7/2 | -1-√-7/2 | 1 | 1 | -1+√-7/2 | -1-√-7/2 | complex lifted from GL3(𝔽2) |
| ρ4 | 4 | -4 | 1 | 0 | -1 | 1+√-7/2 | 1-√-7/2 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ5 | 4 | -4 | 1 | 0 | -1 | 1-√-7/2 | 1+√-7/2 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ6 | 6 | 6 | 0 | 2 | 0 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from GL3(𝔽2) |
| ρ7 | 6 | -6 | 0 | 0 | 0 | -1 | -1 | √2 | -√2 | 1 | 1 | symplectic faithful, Schur index 2 |
| ρ8 | 6 | -6 | 0 | 0 | 0 | -1 | -1 | -√2 | √2 | 1 | 1 | symplectic faithful, Schur index 2 |
| ρ9 | 7 | 7 | 1 | -1 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from GL3(𝔽2) |
| ρ10 | 8 | 8 | -1 | 0 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | orthogonal lifted from GL3(𝔽2) |
| ρ11 | 8 | -8 | -1 | 0 | 1 | 1 | 1 | 0 | 0 | -1 | -1 | symplectic faithful, Schur index 2 |
►On 16 points - transitive group 16T715
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 10 11)(2 4 7)(3 6 8)(5 14 15)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10,11)(2,4,7)(3,6,8)(5,14,15)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10,11)(2,4,7)(3,6,8)(5,14,15) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,10,11),(2,4,7),(3,6,8),(5,14,15)]])
G:=TransitiveGroup(16,715);
Matrix representation of SL2(𝔽7) ►in GL2(𝔽7) generated by
| 5 | 4 |
| 2 | 6 |
| 0 | 4 |
| 5 | 6 |
G:=sub<GL(2,GF(7))| [5,2,4,6],[0,5,4,6] >;
SL2(𝔽7) in GAP, Magma, Sage, TeX
{\rm SL}_2({\mathbb F}_7) % in TeX
G:=Group("SL(2,7)"); // GroupNames label
G:=SmallGroup(336,114);
// by ID
G=gap.SmallGroup(336,114);
# by ID
Export
Subgroup lattice of SL2(𝔽7) in TeX
Character table of SL2(𝔽7) in TeX