Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D45 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C8×D45 |
| Fitting | C5×C8×C9 ⋊ C2 |
| derived, Hall | C5×C9 ⋊ C2×C8 |
| Sylow | C9 ⋊ C8×D5 |
| Sylow | C5 ⋊ C8×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D45 |
| commutator, hypocenter | C5×C9 | C2×C8 |
| Frattini | C3×C4 | C2×D15 |
| Fitting | C5×C8×C9 | C2 |
| socle | C2×C3×C5 | C4×D3 |
| prime radical | C2×D45 | C4 |
| cosocle | C4×C5×C9 | C22 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C8 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
192 conjugacy classes — view the full character table.
| maximal subgroup of | #13083691, #13083693, C16×D45, … |
| maximal normal subgroup of | #13083691, #13083693, C16×D45, … |
| maximal quotient of | C16×D45, #13083841, C8×Dic45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 66);