#13166114

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, hypercenter	C₂²	#89101
commutator	C₂×C₃²×A₄	C₂×C₄
Fitting, socle	C₂⁴×C₃²	C₂×D₃
prime radical	C₂²×C₃×C₃⋊S₄	C₂
cosocle	C₂²×C₃²×A₄	C₂²
hypocenter	C₃²×A₄	C₂²⋊C₄
LCS	G ▷ C₂×C₃²×A₄ ▷ C₃²×A₄
UCS	C₁ ◁ C₂²

754 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

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

μ_t(G) = 144

Polycyclic presentation:

2^4,3^3,2^2|b^2=c,b^a=bd,e^a=e^2,f^a=f^2,h^a=hi,f^b=f^2,g^b=g^2,h^e=i,i^e=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 32177);