Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C9 · Schur multiplier (H2) = C215 |
| direct product | C25×D3×C9 |
| unambiguous semidirect product | C3⋊(C26×C9) |
| unambiguous semidirect product | (C2×C3)⋊(C25×C9) |
| Fitting | C25×C3×C9 ⋊ C2 |
| derived | C3 ⋊ C26×C9 |
| Sylow | C3×C9 ⋊ C26 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25×C9 | D3 |
| commutator, hypocenter | C3 | C26×C9 |
| Frattini | C3 | C25×C3×D3 |
| Fitting | C25×C3×C9 | C2 |
| socle | C25×C32 | C2×C3 |
| prime radical | C25×C3×D3 | C3 |
| cosocle | C32 | C26×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C25×C9 | |
17698 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
864 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46182);