order	162
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂×C₃×AGL(2,𝔽₃) (order 2592) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×C₃×S₄
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₉×C₃⋊D₃
unambiguous semidirect product	C₃⋊(D₃×C₉)

Splittings

Fitting, Sylow	C₃²×C₉ ⋊ C₂
derived	C₃² ⋊ C₂×C₉

52 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 15 points:

(2,3)(5,6)(7,8,10,9,11,13,12,14,15);(4,5,6);(1,2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 39);