Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C3 |
| direct product | C22×#89039 |
| Fitting | C24×C33 ⋊ C22 |
| Sylow | C33 ⋊ C23×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C23×D3×C3⋊D3 |
| Fitting | C24×C33 | C22 |
| socle | C23×C33 | C23 |
| hypercenter | C24 | D3×C3⋊D3 |
| hypocenter | C33 | C23×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ C24 | |
11164 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
204 conjugacy classes — view the full character table.
| maximal subgroup of | C22×((C3⋊S4)⋊C3)⋊D3, #72222930, #72222978, … |
| maximal normal subgroup of | #72222930, #72222978, C22×D3×D4×C3⋊D3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47725);