Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C4 |
| direct product | C4×C5×Dic9 |
| unambiguous semidirect product | C9⋊(C42×C5) |
| derived, Sylow | C9 ⋊ C42×C5 |
| Hall | C5×C9 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C5 | D9 |
| commutator, hypocenter | C9 | C42×C5 |
| Frattini | C22×C3 | C2×C5×D3 |
| Fitting | C2×C4×C5×C9 | C2 |
| socle, prime radical | C22×C3×C5 | C2×D3 |
| cosocle | C22×C9 | C22×C5 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C4×C5 | |
90 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | Dic9×#168, Dic9⋊#168, #13083725, … |
| maximal normal subgroup of | Dic9×#168, Dic9⋊#168, #13083725, … |
| maximal quotient of | C4×C5×C9⋊C8, C5×#63362, C5×C8×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 56);