non-abelian, simple, perfect, not soluble
Aliases: GL3(𝔽2), SL3(𝔽2), PGL3(𝔽2), PSL3(𝔽2), PSL2(𝔽7), PSU2(𝔽7), Ω3(𝔽7), PΩ3(𝔽7), PSL(2,7), Aut(C23), also denoted L3(2) (L=PSL), also denoted L2(7) (L=PSL), SmallGroup(168,42)
Series: Chief►Derived ►Lower central ►Upper central
| C1 — GL3(𝔽2) |
| GL3(𝔽2) |
| GL3(𝔽2) |
Character table of GL3(𝔽2)
| class | 1 | 2 | 3 | 4 | 7A | 7B | |
| size | 1 | 21 | 56 | 42 | 24 | 24 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 3 | -1 | 0 | 1 | -1+√-7/2 | -1-√-7/2 | complex faithful |
| ρ3 | 3 | -1 | 0 | 1 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ4 | 6 | 2 | 0 | 0 | -1 | -1 | orthogonal faithful |
| ρ5 | 7 | -1 | 1 | -1 | 0 | 0 | orthogonal faithful |
| ρ6 | 8 | 0 | -1 | 0 | 1 | 1 | orthogonal faithful |
►On 7 points: primitive, doubly transitive - transitive group 7T5
Generators in S7
(1 2)(4 5)
(2 3 4)(5 6 7)
G:=sub<Sym(7)| (1,2)(4,5), (2,3,4)(5,6,7)>;
G:=Group( (1,2)(4,5), (2,3,4)(5,6,7) );
G=PermutationGroup([[(1,2),(4,5)], [(2,3,4),(5,6,7)]])
G:=TransitiveGroup(7,5);
►On 8 points: primitive, doubly transitive - transitive group 8T37
Generators in S8
(1 3)(2 5)(4 6)(7 8)
(3 4 5)(6 7 8)
G:=sub<Sym(8)| (1,3)(2,5)(4,6)(7,8), (3,4,5)(6,7,8)>;
G:=Group( (1,3)(2,5)(4,6)(7,8), (3,4,5)(6,7,8) );
G=PermutationGroup([[(1,3),(2,5),(4,6),(7,8)], [(3,4,5),(6,7,8)]])
G:=TransitiveGroup(8,37);
►On 14 points - transitive group 14T10
Generators in S14
(1 11)(2 3)(4 9)(5 7)(8 14)(10 13)
(3 4 5)(6 7 8)(9 10 11)(12 13 14)
G:=sub<Sym(14)| (1,11)(2,3)(4,9)(5,7)(8,14)(10,13), (3,4,5)(6,7,8)(9,10,11)(12,13,14)>;
G:=Group( (1,11)(2,3)(4,9)(5,7)(8,14)(10,13), (3,4,5)(6,7,8)(9,10,11)(12,13,14) );
G=PermutationGroup([[(1,11),(2,3),(4,9),(5,7),(8,14),(10,13)], [(3,4,5),(6,7,8),(9,10,11),(12,13,14)]])
G:=TransitiveGroup(14,10);
►On 21 points - transitive group 21T14
Generators in S21
(1 9)(2 11)(3 17)(4 18)(6 19)(7 15)(8 16)(12 20)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)
G:=sub<Sym(21)| (1,9)(2,11)(3,17)(4,18)(6,19)(7,15)(8,16)(12,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)>;
G:=Group( (1,9)(2,11)(3,17)(4,18)(6,19)(7,15)(8,16)(12,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) );
G=PermutationGroup([[(1,9),(2,11),(3,17),(4,18),(6,19),(7,15),(8,16),(12,20)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21)]])
G:=TransitiveGroup(21,14);
►On 24 points - transitive group 24T284
Generators in S24
(1 8)(2 14)(3 19)(4 21)(5 6)(7 10)(9 18)(11 12)(13 23)(15 17)(16 20)(22 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)
G:=sub<Sym(24)| (1,8)(2,14)(3,19)(4,21)(5,6)(7,10)(9,18)(11,12)(13,23)(15,17)(16,20)(22,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)>;
G:=Group( (1,8)(2,14)(3,19)(4,21)(5,6)(7,10)(9,18)(11,12)(13,23)(15,17)(16,20)(22,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24) );
G=PermutationGroup([[(1,8),(2,14),(3,19),(4,21),(5,6),(7,10),(9,18),(11,12),(13,23),(15,17),(16,20),(22,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)]])
G:=TransitiveGroup(24,284);
►On 28 points - transitive group 28T32
Generators in S28
(1 2)(3 5)(4 14)(6 20)(7 25)(8 18)(10 27)(11 23)(13 28)(15 26)(16 22)(19 24)
(2 3 4)(5 6 7)(8 9 10)(11 12 13)(14 15 16)(17 18 19)(20 21 22)(23 24 25)(26 27 28)
G:=sub<Sym(28)| (1,2)(3,5)(4,14)(6,20)(7,25)(8,18)(10,27)(11,23)(13,28)(15,26)(16,22)(19,24), (2,3,4)(5,6,7)(8,9,10)(11,12,13)(14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)>;
G:=Group( (1,2)(3,5)(4,14)(6,20)(7,25)(8,18)(10,27)(11,23)(13,28)(15,26)(16,22)(19,24), (2,3,4)(5,6,7)(8,9,10)(11,12,13)(14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28) );
G=PermutationGroup([[(1,2),(3,5),(4,14),(6,20),(7,25),(8,18),(10,27),(11,23),(13,28),(15,26),(16,22),(19,24)], [(2,3,4),(5,6,7),(8,9,10),(11,12,13),(14,15,16),(17,18,19),(20,21,22),(23,24,25),(26,27,28)]])
G:=TransitiveGroup(28,32);
GL3(𝔽2) is a maximal subgroup of
PGL2(𝔽7)
GL3(𝔽2) is a maximal quotient of SL2(𝔽7)
| action | f(x) | Disc(f) |
|---|---|---|
| 7T5 | x7-2x6+2x4-2x3+2x2-2 | 26·3172 |
| 8T37 | x8-3x7-28x6+63x5+252x4-357x3-728x2+309x+151 | 222·78·974·1272 |
| 14T10 | x14-21x12-35x11+119x10+455x9+371x8-894x7-2905x6-3668x5-2072x4-147x3+133x2-196x-112 | 212·716·176·56310665212 |
Matrix representation of GL3(𝔽2) ►in GL3(𝔽2) generated by
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
G:=sub<GL(3,GF(2))| [0,1,1,1,0,1,0,0,1],[1,0,0,0,0,1,0,1,1] >;
GL3(𝔽2) in GAP, Magma, Sage, TeX
{\rm GL}_3({\mathbb F}_2) % in TeX
G:=Group("GL(3,2)"); // GroupNames label
G:=SmallGroup(168,42);
// by ID
G=gap.SmallGroup(168,42);
# by ID
Export
Subgroup lattice of GL3(𝔽2) in TeX
Character table of GL3(𝔽2) in TeX