#13166026

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×C₃⋊S₄
commutator, prime radical, cosocle	C₂×C₃²×SL(2,𝔽₃)	C₂²
Fitting	C₂×C₃²×Q₈	C₂×D₃
socle	C₂²×C₃²	C₂×S₄
hypocenter	C₃²×SL(2,𝔽₃)	Q₈
LCS	G ▷ C₂×C₃²×SL(2,𝔽₃) ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂²

406 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

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

μ_t(G) = 192

Polycyclic presentation:

2^3,3^3,2^3|a^2=c,b^2=ci,g^2=i,h^2=i,b^a=bci,d^a=d^2,e^a=e^2,g^a=gh,h^a=hi,f^b=f^2,g^d=h,h^d=ghi,h^g=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 32089);