Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C32 · Schur multiplier (H2) = C215×C3 |
| direct product | C25×C32×D3 |
| unambiguous semidirect product | C3⋊(C26×C32) |
| unambiguous semidirect product | (C2×C3)⋊(C25×C32) |
| Fitting, socle | C25×C33 ⋊ C2 |
| derived | C3 ⋊ C26×C32 |
| Sylow | C33 ⋊ C26 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25×C32 | D3 |
| commutator, cosocle, hypocenter | C3 | C26×C32 |
| Frattini | C1 | G |
| Fitting, socle | C25×C33 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C25×C32 | |
36892 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, 47936);