#13165992

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, hypercenter	C₂²	C₃⋊D₃×S₄
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

1114 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1114 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

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

μ_t(G) = 216

Polycyclic presentation:

2^4,3^3,2^2|a^2=d,b^2=cd,b^a=bc,f^a=f^2,g^a=g^2,e^b=e^2,h^b=hi,h^e=i,i^e=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 32055);