Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C3⋊#10593125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D3×C22⋊A4 |
| commutator, hypocenter | C3×H(5,𝔽2) | C2×C9 |
| Fitting, cosocle | C32×H(5,𝔽2) | C2×C3 |
| socle | C2×C32 | C2×C22⋊A4 |
| prime radical | C3×#5966 | C3 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
670 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^2,2^4,3,2|a^2=i,b^3=c,h^a=h^2,d^b=de,e^b=d,f^b=fg,g^b=f,g^d=gi,f^e=fi99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30722);