Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D5×D9 |
| unambiguous semidirect product | (C5×D9)⋊C23 |
| unambiguous semidirect product | (C9×D5)⋊C23 |
| unambiguous semidirect product | D45⋊C23 |
| Fitting | C22×C5×C9 ⋊ C22 |
| derived, Hall | C5×C9 ⋊ C24 |
| Sylow | C9 ⋊ C23×D5 |
| Sylow | C5 ⋊ C23×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D5×D9 |
| commutator, cosocle, hypocenter | C5×C9 | C24 |
| Frattini | C3 | C22×D3×D5 |
| Fitting | C22×C5×C9 | C22 |
| socle | C22×C3×C5 | C2×D3 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C22 | |
402 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | D5×(C2×D9)⋊C4, D9×#685, #13084232, … |
| maximal normal subgroup of | D5×(C2×D9)⋊C4, D9×#685, #13084232, … |
| maximal quotient of | C2×D9×Dic10, C2×#10623657, #13085254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 402);