C₅×C₃⋊S₄ #67210

order	360
isoclinism	isoclinic class of C₃⋊S₄
automorphism	Aut(G) = C₄×((C₃⋊S₄)⋊C₃)⋊C₂ (order 1728) · Inn(G) = C₃⋊S₄ · Out(G) = C₄×D₃
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₅×C₃⋊S₄
unambiguous semidirect product	C₃⋊(C₅×S₄)
unambiguous semidirect product	A₄⋊(C₅×D₃)

Fitting, socle	C₂²×C₃×C₅ ⋊ D₃
derived	C₃×A₄ ⋊ C₂×C₅

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

60 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 140);