C₂×C₅×A₄ #1233

order	120
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₄×S₄ (order 96) · Inn(G) = A₄ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₅)×A₄⋊S₄
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×A₄
unambiguous semidirect product	C₂³⋊(C₃×C₅)
unambiguous semidirect product	(C₂³×C₅)⋊C₃

	subgroup	G/H
center, hypercenter	C₂×C₅	A₄
commutator, cosocle, hypocenter	C₂²	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂³×C₅	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₅

24 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 43);