Copied to
clipboard

G = SL2(𝔽3)  order 24 = 23·3

Special linear group on 𝔽32

non-abelian, soluble

Aliases: SL2(𝔽3), SU2(𝔽3), Spin3(𝔽3), Q8⋊C3, C2.A4, Binary tetrahedral group (2T, <2,3,3>), 1st non-monomial group, SmallGroup(24,3)

Series: Derived Chief Lower central Upper central

C1C2Q8 — SL2(𝔽3)
C1C2Q8 — SL2(𝔽3)
Q8 — SL2(𝔽3)
C1C2

Generators and relations for SL2(𝔽3)
 G = < a,b,c | a4=c3=1, b2=a2, bab-1=a-1, cac-1=b, cbc-1=ab >

4C3
3C4
4C6

Character table of SL2(𝔽3)

 class 123A3B46A6B
 size 1144644
ρ11111111    trivial
ρ211ζ32ζ31ζ32ζ3    linear of order 3
ρ311ζ3ζ321ζ3ζ32    linear of order 3
ρ42-2-1-1011    symplectic faithful, Schur index 2
ρ52-2ζ65ζ60ζ3ζ32    complex faithful
ρ62-2ζ6ζ650ζ32ζ3    complex faithful
ρ73300-100    orthogonal lifted from A4

Permutation representations of SL2(𝔽3)
On 8 points - transitive group 8T12
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 7 3 5)(2 6 4 8)
(2 6 7)(4 8 5)

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

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

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

G:=TransitiveGroup(8,12);

Regular action on 24 points - transitive group 24T7
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 7 3 5)(2 6 4 8)(9 13 11 15)(10 16 12 14)(17 22 19 24)(18 21 20 23)
(1 17 13)(2 21 11)(3 19 15)(4 23 9)(5 20 12)(6 22 14)(7 18 10)(8 24 16)

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), (1,7,3,5)(2,6,4,8)(9,13,11,15)(10,16,12,14)(17,22,19,24)(18,21,20,23), (1,17,13)(2,21,11)(3,19,15)(4,23,9)(5,20,12)(6,22,14)(7,18,10)(8,24,16)>;

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), (1,7,3,5)(2,6,4,8)(9,13,11,15)(10,16,12,14)(17,22,19,24)(18,21,20,23), (1,17,13)(2,21,11)(3,19,15)(4,23,9)(5,20,12)(6,22,14)(7,18,10)(8,24,16) );

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

G:=TransitiveGroup(24,7);

Polynomial with Galois group SL2(𝔽3) over ℚ
actionf(x)Disc(f)
8T12x8-3x7-8x6+24x5+9x4-34x3-4x2+11x-124·2774

Matrix representation of SL2(𝔽3) in GL2(𝔽3) generated by

02
10
,
21
11
,
11
01
G:=sub<GL(2,GF(3))| [0,1,2,0],[2,1,1,1],[1,0,1,1] >;

SL2(𝔽3) in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("SL(2,3)");
// GroupNames label

G:=SmallGroup(24,3);
// by ID

G=gap.SmallGroup(24,3);
# by ID

G:=PCGroup([4,-3,-2,2,-2,49,37,110,78,34]);
// Polycyclic

G:=Group<a,b,c|a^4=c^3=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,c*b*c^-1=a*b>;
// generators/relations

׿
×
𝔽