Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C23×C5×D9 |
| unambiguous semidirect product | C9⋊(C24×C5) |
| unambiguous semidirect product | (C2×C9)⋊(C23×C5) |
| Fitting | C23×C5×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C24×C5 |
| Hall | C5×C9 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C5 | D9 |
| commutator, cosocle, hypocenter | C9 | C24×C5 |
| Frattini | C3 | C23×C5×D3 |
| Fitting | C23×C5×C9 | C2 |
| socle | C23×C3×C5 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C23×C5 | |
402 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C2×(C2×D9)⋊Dic5, D9×C22⋊Dic5, #13084321, … |
| maximal normal subgroup of | C2×(C2×D9)⋊Dic5, D9×C22⋊Dic5, #13084321, … |
| maximal quotient of | C22×C5×Dic18, C22×C4×C5×D9, C22×C5×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 406);