C₃×C₅⋊S₄ #67209

order	360
isoclinism	isoclinic class of C₅⋊S₄
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×S₄ (order 960) · Inn(G) = C₅⋊S₄ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃×C₅⋊S₄
unambiguous semidirect product	C₅⋊(C₃×S₄)
unambiguous semidirect product	A₄⋊(C₃×D₅)

	subgroup	G/H
center, hypercenter	C₃	C₅⋊S₄
commutator, cosocle, hypocenter	C₅×A₄	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅	D₃
LCS	G ▷ C₅×A₄
UCS	C₁ ◁ C₃

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 60

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 139);