C₅×D₃×A₄ #67213

order	360
isoclinism	isoclinic class of D₃×A₄
automorphism	Aut(G) = C₄×D₃×S₄ (order 576) · Inn(G) = D₃×A₄ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₃×A₄
unambiguous semidirect product	C₃⋊(C₂×C₅×A₄)
unambiguous semidirect product	(C₂²×D₃)⋊(C₃×C₅)
unambiguous semidirect product	(C₂²×C₅×D₃)⋊C₃

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

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

52 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 143);