#272⋊C₉ #92179

order	486
isoclinism	isoclinic class of #4174
automorphism	Aut(G) = C₃²×((C₃⋊D₃)⋊C₃)⋊C₂ (order 972) · Inn(G) = (C₃⋊D₃)⋊C₃ · Out(G) = C₂×C₃²
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

Fitting, Sylow	H(3,𝔽₃)⋊C₉ ⋊ C₂
derived	H(3,𝔽₃) ⋊ C₂×C₉

67 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

2,3^5|b^3=c,d^a=d^2,e^a=e^2f,d^b=de,e^d=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 24);