Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C3 |
| direct product | C32×#5609 |
| Fitting | C33×#121 ⋊ C2 |
| Sylow | C33 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C23×D3 |
| commutator, cosocle | C22×C3 | C24×C32 |
| Frattini | C22 | C23×C32×D3 |
| Fitting | C33×#121 | C2 |
| socle | C22×C33 | C24 |
| hypercenter | C32×#121 | D3 |
| hypocenter | C3 | C32×D42 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C32 ◁ C32×#121 | |
2944 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
351 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 45051);