#13166006

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

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×C₃⋊S₄
commutator	C₂×C₃²×A₄	C₂×C₄
Fitting	C₂²×C₃²×C₄	C₂×D₃
socle	C₂³×C₃²	C₂×Dic₃
prime radical	C₂×C₃×D₃×A₄	C₄
cosocle	C₃²×C₄×A₄	C₂²
hypocenter	C₃²×A₄	M₁₆
LCS	G ▷ C₂×C₃²×A₄ ▷ C₃²×A₄
UCS	C₁ ◁ C₄

478 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 76 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|a^2=c,c^2=d,b^a=bd,e^a=e^2,f^a=f^2,h^a=hi,g^b=g^2,h^e=i,i^e=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 32069);