Copied to
clipboard

G = GL2(𝔽3)  order 48 = 24·3

General linear group on 𝔽32

non-abelian, soluble

Aliases: GL2(𝔽3), Q8⋊S3, C2.3S4, SL2(𝔽3)⋊C2, Aut(C32), SmallGroup(48,29)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — GL2(𝔽3)
C1C2Q8SL2(𝔽3) — GL2(𝔽3)
SL2(𝔽3) — GL2(𝔽3)
C1C2

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 12A2B3468A8B
 size 111286866
ρ111111111    trivial
ρ211-1111-1-1    linear of order 2
ρ3220-12-100    orthogonal lifted from S3
ρ42-20-101--2-2    complex faithful
ρ52-20-101-2--2    complex faithful
ρ633-10-1011    orthogonal lifted from S4
ρ73310-10-1-1    orthogonal lifted from S4
ρ84-4010-100    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 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 22 19 24)(18 21 20 23)
(1 15 19)(2 10 23)(3 13 17)(4 12 21)(5 9 20)(6 14 22)(7 11 18)(8 16 24)
(2 7)(4 5)(6 8)(9 21)(10 18)(11 23)(12 20)(13 17)(14 24)(15 19)(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)(17,22,19,24)(18,21,20,23), (1,15,19)(2,10,23)(3,13,17)(4,12,21)(5,9,20)(6,14,22)(7,11,18)(8,16,24), (2,7)(4,5)(6,8)(9,21)(10,18)(11,23)(12,20)(13,17)(14,24)(15,19)(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)(17,22,19,24)(18,21,20,23), (1,15,19)(2,10,23)(3,13,17)(4,12,21)(5,9,20)(6,14,22)(7,11,18)(8,16,24), (2,7)(4,5)(6,8)(9,21)(10,18)(11,23)(12,20)(13,17)(14,24)(15,19)(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),(17,22,19,24),(18,21,20,23)], [(1,15,19),(2,10,23),(3,13,17),(4,12,21),(5,9,20),(6,14,22),(7,11,18),(8,16,24)], [(2,7),(4,5),(6,8),(9,21),(10,18),(11,23),(12,20),(13,17),(14,24),(15,19),(16,22)])

G:=TransitiveGroup(24,22);

GL2(𝔽3) is a maximal subgroup of
Q8.D6  C4.6S4  C4.3S4  C6.6S4  Q8⋊S4  Q82S4  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)

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

21
11
,
11
12
,
12
01
,
20
01
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

׿
×
𝔽