#13165388

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

C₃³ ⋊ #386

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×GO⁺(4,𝔽₂)
commutator	C₂×C₃×C₃⋊D₃	C₂²×C₄
Fitting, socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×#88952	C₂
cosocle	C₂²×C₃×C₃⋊D₃	C₂³
hypocenter	C₃³	#386
LCS	G ▷ C₂×C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₂²

1152 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 90 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=f,b^a=bd,h^a=h^2,i^a=h^2i,c^b=ce,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, 31451);