A₄×SL(2,𝔽₃) #64211

order	288
automorphism	Aut(G) = S₄² (order 576) · Inn(G) = A₄² · Out(G) = C₂²
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	A₄×SL(2,𝔽₃)
unambiguous semidirect product	(C₂²×SL(2,𝔽₃))⋊C₃
unambiguous semidirect product	(Q₈×A₄)⋊C₃
unambiguous semidirect product	(C₂²⋊SL(2,𝔽₃))⋊C₃

C₂²×Q₈ ⋊ C₃²

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄²
commutator, Fitting, cosocle, hypocenter	C₂²×Q₈	C₃²
socle	C₂³	C₃×A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂

59 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

(1,4)(2,3)(6,7,9)(10,12,11);(2,3,4)(5,9,8,11)(6,12,10,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 859);