C₂³×S₄ #5978

order	192
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = S₄×AGL(3,𝔽₂) (order 32256) · Inn(G) = S₄ · Out(G) = AGL(3,𝔽₂)
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product	C₂³×S₄
unambiguous semidirect product	A₄⋊C₂⁴
unambiguous semidirect product	(C₂×A₄)⋊C₂³

Fitting, socle	C₂⁵ ⋊ D₃
derived	A₄ ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂³	S₄
commutator, cosocle, hypocenter	A₄	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂⁵	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂³

701 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 4 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1537);