Aliases: GL2(𝔽3), Q8⋊S3, C2.3S4, SL2(𝔽3)⋊C2, Aut(C32), SmallGroup(48,29)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — GL2(𝔽3) |
Generators and relations for GL2(𝔽3)
G = < a,b,c,d | a4=c3=d2=1, b2=a2, bab-1=dbd=a-1, cac-1=ab, dad=a2b, cbc-1=a, dcd=c-1 >
Character table of GL2(𝔽3)
| class | 1 | 2A | 2B | 3 | 4 | 6 | 8A | 8B | |
| size | 1 | 1 | 12 | 8 | 6 | 8 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 2 | -2 | 0 | -1 | 0 | 1 | -√-2 | √-2 | complex faithful |
| ρ5 | 2 | -2 | 0 | -1 | 0 | 1 | √-2 | -√-2 | complex faithful |
| ρ6 | 3 | 3 | -1 | 0 | -1 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ7 | 3 | 3 | 1 | 0 | -1 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ8 | 4 | -4 | 0 | 1 | 0 | -1 | 0 | 0 | orthogonal faithful |
►On 8 points - transitive group 8T23
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 5 3 7)(2 8 4 6)
(2 5 8)(4 7 6)
(1 3)(2 5)(4 7)
G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,5,3,7)(2,8,4,6), (2,5,8)(4,7,6), (1,3)(2,5)(4,7)>;
G:=Group( (1,2,3,4)(5,6,7,8), (1,5,3,7)(2,8,4,6), (2,5,8)(4,7,6), (1,3)(2,5)(4,7) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,5,3,7),(2,8,4,6)], [(2,5,8),(4,7,6)], [(1,3),(2,5),(4,7)]])
G:=TransitiveGroup(8,23);
►On 16 points - transitive group 16T66
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5 3 7)(2 8 4 6)(9 14 11 16)(10 13 12 15)
(2 5 8)(4 7 6)(9 15 14)(11 13 16)
(1 12)(2 13)(3 10)(4 15)(5 11)(6 14)(7 9)(8 16)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5,3,7)(2,8,4,6)(9,14,11,16)(10,13,12,15), (2,5,8)(4,7,6)(9,15,14)(11,13,16), (1,12)(2,13)(3,10)(4,15)(5,11)(6,14)(7,9)(8,16)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5,3,7)(2,8,4,6)(9,14,11,16)(10,13,12,15), (2,5,8)(4,7,6)(9,15,14)(11,13,16), (1,12)(2,13)(3,10)(4,15)(5,11)(6,14)(7,9)(8,16) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,5,3,7),(2,8,4,6),(9,14,11,16),(10,13,12,15)], [(2,5,8),(4,7,6),(9,15,14),(11,13,16)], [(1,12),(2,13),(3,10),(4,15),(5,11),(6,14),(7,9),(8,16)]])
G:=TransitiveGroup(16,66);
►On 24 points - transitive group 24T22
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 5 3 7)(2 8 4 6)(9 14 11 16)(10 13 12 15)(17 24 19 22)(18 23 20 21)
(1 15 17)(2 10 23)(3 13 19)(4 12 21)(5 9 18)(6 14 22)(7 11 20)(8 16 24)
(2 7)(4 5)(6 8)(9 21)(10 20)(11 23)(12 18)(13 19)(14 24)(15 17)(16 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), (1,5,3,7)(2,8,4,6)(9,14,11,16)(10,13,12,15)(17,24,19,22)(18,23,20,21), (1,15,17)(2,10,23)(3,13,19)(4,12,21)(5,9,18)(6,14,22)(7,11,20)(8,16,24), (2,7)(4,5)(6,8)(9,21)(10,20)(11,23)(12,18)(13,19)(14,24)(15,17)(16,22)>;
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), (1,5,3,7)(2,8,4,6)(9,14,11,16)(10,13,12,15)(17,24,19,22)(18,23,20,21), (1,15,17)(2,10,23)(3,13,19)(4,12,21)(5,9,18)(6,14,22)(7,11,20)(8,16,24), (2,7)(4,5)(6,8)(9,21)(10,20)(11,23)(12,18)(13,19)(14,24)(15,17)(16,22) );
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)], [(1,5,3,7),(2,8,4,6),(9,14,11,16),(10,13,12,15),(17,24,19,22),(18,23,20,21)], [(1,15,17),(2,10,23),(3,13,19),(4,12,21),(5,9,18),(6,14,22),(7,11,20),(8,16,24)], [(2,7),(4,5),(6,8),(9,21),(10,20),(11,23),(12,18),(13,19),(14,24),(15,17),(16,22)]])
G:=TransitiveGroup(24,22);
GL2(𝔽3) is a maximal subgroup of
Q8.D6 C4.6S4 C4.3S4 C6.6S4 Q8⋊S4 Q8⋊2S4 C2.S5 Q8⋊D15 Q8⋊D21 AGL2(𝔽3)
GL2(𝔽3) is a maximal quotient of
Q8⋊Dic3 Q8⋊D9 C6.6S4 C23.8S4 Q8⋊S4 Q8⋊D15 Q8⋊D21 AGL2(𝔽3)
| action | f(x) | Disc(f) |
|---|---|---|
| 8T23 | x8-4x7-4x6+26x5+2x4-52x3+31x+1 | 27773 |
Matrix representation of GL2(𝔽3) ►in GL2(𝔽3) generated by
| 2 | 1 |
| 1 | 1 |
| 1 | 1 |
| 1 | 2 |
| 1 | 2 |
| 0 | 1 |
| 2 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(3))| [2,1,1,1],[1,1,1,2],[1,0,2,1],[2,0,0,1] >;
GL2(𝔽3) in GAP, Magma, Sage, TeX
{\rm GL}_2({\mathbb F}_3) % in TeX
G:=Group("GL(2,3)"); // GroupNames label
G:=SmallGroup(48,29);
// by ID
G=gap.SmallGroup(48,29);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-2,41,182,277,72,123,188,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^3=d^2=1,b^2=a^2,b*a*b^-1=d*b*d=a^-1,c*a*c^-1=a*b,d*a*d=a^2*b,c*b*c^-1=a,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of GL2(𝔽3) in TeX
Character table of GL2(𝔽3) in TeX