Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C3 |
| direct product | C2×#11724611 |
| Fitting | C24×C33 ⋊ C22 |
| Sylow | C33 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C23×D3×C3⋊D3 |
| Fitting | C24×C33 | C22 |
| socle | C22×C33 | C24 |
| hypercenter | C24 | D3×C3⋊D3 |
| hypocenter | C33 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C24 | |
6332 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47726);