Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#11724420 |
| Sylow | C33 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D33 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C22×D33 |
| Fitting | C23×C33 | C23 |
| socle | C22×C33 | C24 |
| hypercenter | C23 | D33 |
| hypocenter | C33 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
4434 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47390);