A₄⋊#272 #10620056

order	648
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂×C₃²

Aliases

Fitting	C₂²×H(3,𝔽₃) ⋊ D₃
derived	A₄×H(3,𝔽₃) ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₃⋊(C₃⋊S₄)
commutator, cosocle, hypocenter	A₄×H(3,𝔽₃)	C₂
Fitting	C₂²×H(3,𝔽₃)	D₃
socle	C₂²×C₃	C₃⋊(C₃⋊D₃)
LCS	G ▷ A₄×H(3,𝔽₃)
UCS	C₁ ◁ C₃

227 conjugacy classes of subgroups.

μ_t(G) = 36

Polycyclic presentation:

2,3^4,2^2|b^a=b^2,c^a=c^2,d^a=d^2,f^a=g,g^a=f,f^b=g,g^b=fg,d^c=de

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 575);