C₂×C₃×C₅×D₃ #4388

order	180
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×C₄×D₃ (order 96) · Inn(G) = D₃ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂²×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2)(4,5)(6,7,8)(9,10,11,12,13);(1,2)(3,4,5)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 33);