order	81
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#4208 (order 324) · Inn(G) = C₁ · Out(G) = C₂×#4208
holomorph	G ⋊ Aut(G) = #72221873
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

C₃×C₂₇

Subgroups

14 conjugacy classes of subgroups.

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27);(28,29,30)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(81, 5);