Aliases: SL2(𝔽3), SU2(𝔽3), Spin3(𝔽3), Q8⋊C3, C2.A4, Binary tetrahedral group (2T, <2,3,3>), 1st non-monomial group, SmallGroup(24,3)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — SL2(𝔽3) |
Generators and relations for SL2(𝔽3)
G = < a,b,c | a4=c3=1, b2=a2, bab-1=a-1, cac-1=b, cbc-1=ab >
Character table of SL2(𝔽3)
| class | 1 | 2 | 3A | 3B | 4 | 6A | 6B | |
| size | 1 | 1 | 4 | 4 | 6 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 2 | -2 | -1 | -1 | 0 | 1 | 1 | symplectic faithful, Schur index 2 |
| ρ5 | 2 | -2 | ζ65 | ζ6 | 0 | ζ3 | ζ32 | complex faithful |
| ρ6 | 2 | -2 | ζ6 | ζ65 | 0 | ζ32 | ζ3 | complex faithful |
| ρ7 | 3 | 3 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from A4 |
►On 8 points - transitive group 8T12
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 7 3 5)(2 6 4 8)
(2 6 7)(4 8 5)
G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,7,3,5)(2,6,4,8), (2,6,7)(4,8,5)>;
G:=Group( (1,2,3,4)(5,6,7,8), (1,7,3,5)(2,6,4,8), (2,6,7)(4,8,5) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,7,3,5),(2,6,4,8)], [(2,6,7),(4,8,5)]])
G:=TransitiveGroup(8,12);
►Regular action on 24 points - transitive group 24T7
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 3 5)(2 6 4 8)(9 13 11 15)(10 16 12 14)(17 22 19 24)(18 21 20 23)
(1 17 13)(2 21 11)(3 19 15)(4 23 9)(5 20 12)(6 22 14)(7 18 10)(8 24 16)
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), (1,7,3,5)(2,6,4,8)(9,13,11,15)(10,16,12,14)(17,22,19,24)(18,21,20,23), (1,17,13)(2,21,11)(3,19,15)(4,23,9)(5,20,12)(6,22,14)(7,18,10)(8,24,16)>;
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), (1,7,3,5)(2,6,4,8)(9,13,11,15)(10,16,12,14)(17,22,19,24)(18,21,20,23), (1,17,13)(2,21,11)(3,19,15)(4,23,9)(5,20,12)(6,22,14)(7,18,10)(8,24,16) );
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)], [(1,7,3,5),(2,6,4,8),(9,13,11,15),(10,16,12,14),(17,22,19,24),(18,21,20,23)], [(1,17,13),(2,21,11),(3,19,15),(4,23,9),(5,20,12),(6,22,14),(7,18,10),(8,24,16)]])
G:=TransitiveGroup(24,7);
SL2(𝔽3) is a maximal subgroup of
CSU2(𝔽3) GL2(𝔽3) C4.A4 Q8⋊A4 C23⋊A4 SL2(𝔽5) C14.A4 ASL2(𝔽3) C26.A4 C38.A4
SL2(𝔽3) is a maximal quotient of
Q8⋊C9 C23.3A4 Q8⋊A4 C14.A4 ASL2(𝔽3) C26.A4 C38.A4
| action | f(x) | Disc(f) |
|---|---|---|
| 8T12 | x8-3x7-8x6+24x5+9x4-34x3-4x2+11x-1 | 24·2774 |
Matrix representation of SL2(𝔽3) ►in GL2(𝔽3) generated by
| 0 | 2 |
| 1 | 0 |
| 2 | 1 |
| 1 | 1 |
| 1 | 1 |
| 0 | 1 |
G:=sub<GL(2,GF(3))| [0,1,2,0],[2,1,1,1],[1,0,1,1] >;
SL2(𝔽3) in GAP, Magma, Sage, TeX
{\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("SL(2,3)"); // GroupNames label
G:=SmallGroup(24,3);
// by ID
G=gap.SmallGroup(24,3);
# by ID
G:=PCGroup([4,-3,-2,2,-2,49,37,110,78,34]);
// Polycyclic
G:=Group<a,b,c|a^4=c^3=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,c*b*c^-1=a*b>;
// generators/relations
Export
Subgroup lattice of SL2(𝔽3) in TeX
Character table of SL2(𝔽3) in TeX