order	162
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	Aut(G) = C₃×#4183 (order 486) · Inn(G) = C₃² · Out(G) = #272
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₂×C₉⋊C₉
unambiguous semidirect product	(C₂×C₉)⋊C₉

Subgroups

34 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2)(4,6,9)(7,11,10)(12,13,15,14,16,18,17,19,20);(3,4,7,5,6,10,8,9,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 25);