C₂×C₃⋊#272 #66458

order	324
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product	C₂×C₃⋊#272
unambiguous semidirect product	(C₂×C₃)⋊#272

Fitting	C₂×C₃×H(3,𝔽₃) ⋊ C₂
derived, Sylow	C₃×H(3,𝔽₃) ⋊ C₂²

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

240 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 150);