#13136730

order	1728
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃×C₄	PSU(3,𝔽₂)
commutator	#272	C₄×C₈
Fitting	C₂×C₄×H(3,𝔽₃)	Q₈
socle	C₂²×C₃	#3777
prime radical	C₂²×#272	C₂×C₄
cosocle	C₂×C₄×#272	C₂²
hypocenter	H(3,𝔽₃)	#336
LCS	G ▷ #272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₄

231 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

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

μ_t(G) = 288

Polycyclic presentation:

2^6,3^3|a^2=c,b^2=ce,c^2=f,d^2=f,b^a=bcdf,g^a=g^2h^2i,h^a=g^2h,g^b=gh^2i^2,h^b=g^2h^2,g^c=g^2i,h^c=h^2i^2,h^g=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 2793);