C₃²×D₃² #66473

order	324
isoclinism	isoclinic class of D₃²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃²×D₃²
unambiguous semidirect product	(C₃×D₃)⋊(C₂×C₃²)

Fitting, socle, Sylow	C₃⁴ ⋊ C₂²
derived	C₃² ⋊ C₂²×C₃²

	subgroup	G/H
center, hypercenter	C₃²	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₃⁴	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃²

224 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 165);