Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#672 |
| Fitting | C24×C33 ⋊ C22 |
| Sylow | C33 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C22×D32 |
| commutator, cosocle | C22×C32 | C24×C3 |
| Frattini | C22 | C22×C3×D32 |
| Fitting | C24×C33 | C22 |
| socle | C22×C33 | C24 |
| hypercenter | C24×C3 | D32 |
| hypocenter | C32 | C3×D42 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 | |
2448 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
243 conjugacy classes — view the full character table.
| maximal subgroup of | #67×((C22⋊C9)⋊C2)⋊C3, #67×(C3⋊S4)⋊C3, #72222978 |
| maximal normal subgroup of | #72222978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 41185);