SL(2,𝔽₃) #62

order	24
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = S₄ (order 24) · Inn(G) = A₄ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = SO⁺(4,𝔽₃)
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

special linear	SL(2,𝔽₃)
symplectic	Sp(2,𝔽₃)
special unitary	SU(2,𝔽₃)
unambiguous semidirect product	Q₈⋊C₃

Q₈ ⋊ C₃

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	A₄
commutator, Fitting, cosocle, hypocenter	Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂

7 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

(2,3,5)(6,8,7);(1,5,4,7)(2,8,6,3)

Polycyclic presentation:

3,2^3|b^2=d,c^2=d,b^a=c,c^a=bc,c^b=cd

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24, 3);