order	324
isoclinism	isoclinic class of D₃×(C₃⋊D₃)⋊C₃
automorphism	Aut(G) = D₃×((C₃⋊D₃)⋊C₃)⋊C₂ (order 648) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

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

Splittings

Fitting, Sylow	C₃×H(3,𝔽₃) ⋊ C₂²
derived	C₃³ ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³	C₂²×C₃
Frattini	C₃	C₃×D₃²
Fitting	C₃×H(3,𝔽₃)	C₂²
socle	C₃²	C₂×C₃×D₃
LCS	G ▷ C₃³
UCS	C₁

100 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 18

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 116);