#13135221

order	1728
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂²	#88450
commutator	C₂²×H(3,𝔽₃)	C₄²
Frattini	C₂⁴×C₃	D₃²
Fitting, prime radical, cosocle	C₂⁴×H(3,𝔽₃)	C₂²
socle	C₂²×C₃	(C₂×D₃)⋊Dic₃
hypercenter	C₂⁴	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	#342
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂⁴

629 conjugacy classes of subgroups.

μ_t(G) = 144

Polycyclic presentation:

2^6,3^3|a^2=e,b^2=c,b^a=bd,c^a=cf,h^a=h^2,i^a=i^2,d^b=df,g^b=g^2,i^b=i^2,h^g=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 1284);