Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9×C16 · Schur multiplier (H2) = C23 |
| direct product | C22×C9×C3⋊C16 |
| derived | C3 ⋊ C22×C9×C16 |
| Sylow | C3×C9 ⋊ C22×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8×C9 | D3 |
| commutator, hypocenter | C3 | C22×C9×C16 |
| Frattini | C3×C8 | C22×C3×D3 |
| Fitting | C22×C3×C8×C9 | C2 |
| socle, prime radical | C23×C32 | C3×C8 |
| cosocle | C32×C8 | C23×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C8×C9 | |
370 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, 9133);