#13145677

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

Aliases

No aliases for this group.

C₃³ ⋊ #352

	subgroup	G/H
center	C₂	#11722064
commutator	C₂³×C₃³	C₂×C₄
Frattini, hypercenter	C₂×D₄	D₃×C₃⋊D₃
Fitting, prime radical, cosocle	C₂×C₃³×D₄	C₂²
socle	C₂×C₃³	#99
hypocenter	C₃³	#352
LCS	G ▷ C₂³×C₃³ ▷ C₂²×C₃³ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄

720 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 96 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:

2^6,3^3|a^2=d,b^2=cd,d^2=f,b^a=bc,c^a=ce,e^a=ef,i^a=i^2,c^b=ce,d^b=de,e^b=ef,g^b=g^2,h^b=h^2,d^c=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11740);