Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | D33×D4 |
| Fitting | C33×D4 ⋊ C23 |
| Sylow | C33 ⋊ C23×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D33 |
| commutator, socle, cosocle | C2×C33 | C25 |
| Fitting | C33×D4 | C23 |
| hypercenter | D4 | D33 |
| hypocenter | C33 | C23×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ D4 | |
6210 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | #72220351, D3×#10596675, D33×S4 |
| maximal normal subgroup of | D3×#10596675 |
| maximal quotient of | #72220351, D3×#10595278 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47341);