order	324
isoclinism	isoclinic class of D₃×C₃⋊(C₃⋊D₃)
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product	D₃×C₃⋊(C₃⋊D₃)
unambiguous semidirect product	(C₃×D₃)⋊(C₃⋊D₃)

Splittings

C₃⁴ ⋊ C₂²

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

436 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 4 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 168);