C₂×C₃×(C₃⋊D₃)⋊C₃ #66446

order	324
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	Aut(G) = C₂×D₃×((C₃⋊D₃)⋊C₃)⋊C₂ (order 1296) · Inn(G) = (C₃⋊D₃)⋊C₃ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃²

Aliases

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

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

136 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 138);