order	189
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₉ (order 108) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₉
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×AGL(1,ℤ/27)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₈₉
abelian	C₇×C₂₇

Subgroups

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 189

A minimal faithful permutation representation — 1 generator on 34 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)(28,29,30,31,32,33,34)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(189, 2);