C₂×C₅×S₄ #6835

order	240
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂×C₄×S₄ (order 192) · Inn(G) = S₄ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×C₅×S₄
unambiguous semidirect product	A₄⋊(C₂²×C₅)
unambiguous semidirect product	(C₂×A₄)⋊(C₂×C₅)

Fitting, socle	C₂³×C₅ ⋊ D₃
derived	A₄ ⋊ C₂²×C₅

	subgroup	G/H
center, hypercenter	C₂×C₅	S₄
commutator, cosocle, hypocenter	A₄	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂³×C₅	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂×C₅

66 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 196);