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## G = SL2(𝔽5)  order 120 = 23·3·5

### Special linear group on 𝔽52

non-abelian, perfect, quasisimple, not soluble

Aliases: SL2(𝔽5), SU2(𝔽5), Spin3(𝔽5), C2.A5, Binary icosahedral group (2I or <2,3,5>), SmallGroup(120,5)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — SL2(𝔽5)
 Derived series SL2(𝔽5)
 Lower central SL2(𝔽5)
 Upper central C1 — C2

10C3
6C5
15C4
10C6
6C10
5Q8
10Dic3
6Dic5

Character table of SL2(𝔽5)

 class 1 2 3 4 5A 5B 6 10A 10B size 1 1 20 30 12 12 20 12 12 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 2 -2 -1 0 -1+√5/2 -1-√5/2 1 1+√5/2 1-√5/2 symplectic faithful, Schur index 2 ρ3 2 -2 -1 0 -1-√5/2 -1+√5/2 1 1-√5/2 1+√5/2 symplectic faithful, Schur index 2 ρ4 3 3 0 -1 1-√5/2 1+√5/2 0 1+√5/2 1-√5/2 orthogonal lifted from A5 ρ5 3 3 0 -1 1+√5/2 1-√5/2 0 1-√5/2 1+√5/2 orthogonal lifted from A5 ρ6 4 4 1 0 -1 -1 1 -1 -1 orthogonal lifted from A5 ρ7 4 -4 1 0 -1 -1 -1 1 1 symplectic faithful, Schur index 2 ρ8 5 5 -1 1 0 0 -1 0 0 orthogonal lifted from A5 ρ9 6 -6 0 0 1 1 0 -1 -1 symplectic faithful, Schur index 2

Permutation representations of SL2(𝔽5)
On 24 points - transitive group 24T201
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 14 19)(2 10 18)(3 16 17)(4 12 20)(5 6 11)(7 8 9)(13 24 21)(15 22 23)

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

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

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

G:=TransitiveGroup(24,201);

SL2(𝔽5) is a maximal subgroup of   CSU2(𝔽5)  C2.S5  C4.A5

Matrix representation of SL2(𝔽5) in GL2(𝔽5) generated by

 4 2 4 1
,
 3 3 4 1
G:=sub<GL(2,GF(5))| [4,4,2,1],[3,4,3,1] >;

SL2(𝔽5) in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_5)
% in TeX

G:=Group("SL(2,5)");
// GroupNames label

G:=SmallGroup(120,5);
// by ID

G=gap.SmallGroup(120,5);
# by ID

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