#13165490

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

C₃³ ⋊ #428

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×GO⁺(4,𝔽₂)
commutator	C₃×C₃⋊Dic₃	C₂²×C₄
Fitting, socle	C₂²×C₃³	C₂×D₄
prime radical	C₂²×C₃⋊(C₃⋊D₃)	D₄
cosocle	C₂×C₃×C₃⋊Dic₃	C₂³
hypocenter	C₃³	#428
LCS	G ▷ C₃×C₃⋊Dic₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂²

492 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

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

μ_t(G) = 96

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 31553);