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## G = GL2(𝔽3)  order 48 = 24·3

### General linear group on 𝔽32

Aliases: GL2(𝔽3), Q8⋊S3, C2.3S4, SL2(𝔽3)⋊C2, Aut(C32), Binary octahedral group (2O, <2,3,4>), SmallGroup(48,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3)
 Lower central SL2(𝔽3) — GL2(𝔽3)
 Upper central C1 — C2

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 >

12C2
4C3
3C4
6C22
4S3
4C6
4S3
3D4
3C8
4D6
3SD16

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

Permutation representations of GL2(𝔽3)
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 16 11 14)(10 15 12 13)
(2 5 8)(4 7 6)(9 13 16)(11 15 14)
(1 10)(2 13)(3 12)(4 15)(5 9)(6 14)(7 11)(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)(9,16,11,14)(10,15,12,13), (2,5,8)(4,7,6)(9,13,16)(11,15,14), (1,10)(2,13)(3,12)(4,15)(5,9)(6,14)(7,11)(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)(9,16,11,14)(10,15,12,13), (2,5,8)(4,7,6)(9,13,16)(11,15,14), (1,10)(2,13)(3,12)(4,15)(5,9)(6,14)(7,11)(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),(9,16,11,14),(10,15,12,13)], [(2,5,8),(4,7,6),(9,13,16),(11,15,14)], [(1,10),(2,13),(3,12),(4,15),(5,9),(6,14),(7,11),(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);

Polynomial with Galois group GL2(𝔽3) over ℚ
actionf(x)Disc(f)
8T23x8-4x7-4x6+26x5+2x4-52x3+31x+127773

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

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