Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C23×D3×Dic9 |
| derived, Sylow | C3×C9 ⋊ C24×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D3×D9 |
| commutator, hypocenter | C3×C9 | C24×C4 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×C3×C9 | C22 |
| socle | C24×C32 | C2×D3 |
| prime radical | C24×C3×D3 | D3 |
| cosocle | C2×C3×C9 | C25 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C24 | |
4322 conjugacy classes of subgroups; only the 1681 normal subgroups are shown.
This group has 4322 conjugacy classes of subgroups — too many to draw. Only its 1681 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30866);