C₃²×D₉ #4196

order	162
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = GL(2,𝔽₃)×AGL(1,ℤ/9) (order 2592) · Inn(G) = D₉ · Out(G) = C₃×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

C₃²×D₉

Fitting, Sylow	C₃²×C₉ ⋊ C₂
derived	C₉ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃²	D₉
commutator, cosocle, hypocenter	C₉	C₂×C₃²
Frattini	C₃	C₃²×D₃
Fitting	C₃²×C₉	C₂
socle	C₃³	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₃²

52 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 32);