C₃²×C₁₉ #4298

order	171
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₉×GL(2,𝔽₃) (order 864) · Inn(G) = C₁ · Out(G) = C₂×C₉×GL(2,𝔽₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

C₃²×C₁₉

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 171

A minimal faithful permutation representation — 2 generators on 25 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(171, 5);