#13146565

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂²	C₂×#1088⋊C₂
commutator, prime radical, cosocle	C₂²×#272	C₂³
Frattini, socle	C₂²×C₃	C₂×GO⁺(4,𝔽₂)
Fitting	C₂×C₄×H(3,𝔽₃)	D₄
hypercenter	C₂×C₄	#1088⋊C₂
hypocenter	H(3,𝔽₃)	#400
LCS	G ▷ C₂²×#272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄

385 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

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

μ_t(G) = 288

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 12628);