order	270
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	Aut(G) = C₄×((C₃⋊D₃)⋊C₃)⋊C₂ (order 432) · Inn(G) = (C₃⋊D₃)⋊C₃ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₃

Aliases

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

Splittings

	subgroup	G/H
center, hypercenter	C₅	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	C₃²	C₂×C₃×C₅
Frattini	C₃	C₃×C₅×D₃
Fitting	C₅×H(3,𝔽₃)	C₂
socle	C₃×C₅	C₃×D₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 14 points:

(1,2,3,4,5)(7,12,14,9,10,13)(8,11);(6,7,9)(8,10,13)(11,14,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 10);