C₂×#11722372 #13164460

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

	subgroup	G/H
center	C₂²	C₂×C₉⋊S₄
commutator, cosocle, hypocenter	C₉×SL(2,𝔽₃)	C₂³
Frattini	C₂×C₃	C₂²×C₃⋊S₄
Fitting	C₂×C₉×#41	D₃
socle	C₂²×C₃	C₂×C₃⋊S₄
prime radical	C₂×C₃×#41⋊C₃	D₃
hypercenter	C₂×C₄	C₉⋊S₄
LCS	G ▷ C₉×SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄

563 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

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

μ_t(G) = 576

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 30523);