Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C5×C9⋊C8 |
| unambiguous semidirect product | (C2×C9)⋊(C5×C8) |
| derived, Sylow | C9 ⋊ C2×C5×C8 |
| Hall | C5×C9 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C5 | D9 |
| commutator, hypocenter | C9 | C2×C5×C8 |
| Frattini | C3×C4 | C2×C5×D3 |
| Fitting | C2×C4×C5×C9 | C2 |
| socle, prime radical | C22×C3×C5 | Dic3 |
| cosocle | C4×C9 | C22×C5 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C4×C5 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×C9⋊C8, #13083709, #13083710, … |
| maximal normal subgroup of | Dic5×C9⋊C8, #13083709, #13083710, … |
| maximal quotient of | C4×C5×C9⋊C8, C5×C4⋊(C9⋊C8), C2×C5×C9⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 54);