Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#88668 |
| Fitting | C24×C3×C9 ⋊ C22 |
| Sylow | C3×C9 ⋊ C23×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D3×D9 |
| commutator, cosocle | C2×C3×C9 | C25 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×C3×C9 | C22 |
| socle | C23×C32 | C22×D3 |
| hypercenter | C24 | D3×D9 |
| hypocenter | C3×C9 | C23×D4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C23 ◁ C24 | |
4834 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
204 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13164733 |
| maximal normal subgroup of | C2×#13164733 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30872);