order	180
isoclinism	isoclinic class of D₅×C₃⋊D₃
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(2,𝔽₃) (order 8640) · Inn(G) = G · Out(G) = C₂×S₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	D₅×C₃⋊D₃
unambiguous semidirect product	(C₃×D₅)⋊D₃

Splittings

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

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 3 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 27);