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G = GL3(𝔽2)  order 168 = 23·3·7

General linear group on 𝔽23

non-abelian, simple, perfect, not soluble

Aliases: GL3(𝔽2), SL3(𝔽2), PGL3(𝔽2), PSL3(𝔽2), PSL2(𝔽7), PSU2(𝔽7), Ω3(𝔽7), 3(𝔽7), PSL(2,7), Aut(C23), also denoted L3(2) (L=PSL), also denoted L2(7) (L=PSL), SmallGroup(168,42)

Series: ChiefDerived Lower central Upper central

C1 — GL3(𝔽2)
GL3(𝔽2)
GL3(𝔽2)
C1

21C2
28C3
8C7
7C22
7C22
21C4
28S3
8C7⋊C3
21D4
7A4
7A4
7S4
7S4

Character table of GL3(𝔽2)

 class 12347A7B
 size 12156422424
ρ1111111    trivial
ρ23-101-1+-7/2-1--7/2    complex faithful
ρ33-101-1--7/2-1+-7/2    complex faithful
ρ46200-1-1    orthogonal faithful
ρ57-11-100    orthogonal faithful
ρ680-1011    orthogonal faithful

Permutation representations of GL3(𝔽2)
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 8)(2 6)(3 4)(5 7)
(3 4 5)(6 7 8)

G:=sub<Sym(8)| (1,8)(2,6)(3,4)(5,7), (3,4,5)(6,7,8)>;

G:=Group( (1,8)(2,6)(3,4)(5,7), (3,4,5)(6,7,8) );

G=PermutationGroup([(1,8),(2,6),(3,4),(5,7)], [(3,4,5),(6,7,8)])

G:=TransitiveGroup(8,37);

On 14 points - transitive group 14T10
Generators in S14
(1 10)(2 3)(4 11)(5 14)(6 12)(8 9)
(3 4 5)(6 7 8)(9 10 11)(12 13 14)

G:=sub<Sym(14)| (1,10)(2,3)(4,11)(5,14)(6,12)(8,9), (3,4,5)(6,7,8)(9,10,11)(12,13,14)>;

G:=Group( (1,10)(2,3)(4,11)(5,14)(6,12)(8,9), (3,4,5)(6,7,8)(9,10,11)(12,13,14) );

G=PermutationGroup([(1,10),(2,3),(4,11),(5,14),(6,12),(8,9)], [(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 11)(3 13)(6 20)(7 21)(8 14)(9 17)(10 16)(15 19)
(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)| (1,11)(3,13)(6,20)(7,21)(8,14)(9,17)(10,16)(15,19), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)>;

G:=Group( (1,11)(3,13)(6,20)(7,21)(8,14)(9,17)(10,16)(15,19), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) );

G=PermutationGroup([(1,11),(3,13),(6,20),(7,21),(8,14),(9,17),(10,16),(15,19)], [(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 11)(2 3)(4 5)(6 20)(7 17)(8 21)(9 12)(10 13)(14 19)(15 18)(16 23)(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)| (1,11)(2,3)(4,5)(6,20)(7,17)(8,21)(9,12)(10,13)(14,19)(15,18)(16,23)(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( (1,11)(2,3)(4,5)(6,20)(7,17)(8,21)(9,12)(10,13)(14,19)(15,18)(16,23)(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([(1,11),(2,3),(4,5),(6,20),(7,17),(8,21),(9,12),(10,13),(14,19),(15,18),(16,23),(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 22)(3 10)(4 25)(5 21)(6 15)(7 12)(9 16)(13 18)(14 27)(17 20)(19 23)(24 28)
(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)| (1,22)(3,10)(4,25)(5,21)(6,15)(7,12)(9,16)(13,18)(14,27)(17,20)(19,23)(24,28), (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( (1,22)(3,10)(4,25)(5,21)(6,15)(7,12)(9,16)(13,18)(14,27)(17,20)(19,23)(24,28), (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([(1,22),(3,10),(4,25),(5,21),(6,15),(7,12),(9,16),(13,18),(14,27),(17,20),(19,23),(24,28)], [(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)

Polynomial with Galois group GL3(𝔽2) over ℚ
actionf(x)Disc(f)
7T5x7-2x6+2x4-2x3+2x2-226·3172
8T37x8-3x7-28x6+63x5+252x4-357x3-728x2+309x+151222·78·974·1272
14T10x14-21x12-35x11+119x10+455x9+371x8-894x7-2905x6-3668x5-2072x4-147x3+133x2-196x-112212·716·176·56310665212

Matrix representation of GL3(𝔽2) in GL3(𝔽2) generated by

010
100
111
,
100
001
011
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

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