order	90
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₄×AGL(2,𝔽₃) (order 1728) · Inn(G) = C₃⋊D₃ · Out(G) = C₄×S₄
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₃

Aliases

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

Splittings

Fitting, socle, Hall	C₃²×C₅ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₅

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

24 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(90, 8);