C₅×A₄ #308

order	60
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₄×S₄ (order 96) · Inn(G) = A₄ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×A₄⋊S₄
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×A₄
unambiguous semidirect product	C₂²⋊(C₃×C₅)
unambiguous semidirect product	(C₂²×C₅)⋊C₃

Fitting, socle, Hall	C₂²×C₅ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₅

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

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(60, 9);