Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D45 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C4×D45 |
| Fitting | C2×C4×C5×C9 ⋊ C2 |
| derived, Hall | C5×C9 ⋊ C22×C4 |
| Sylow | C9 ⋊ C2×C4×D5 |
| Sylow | C5 ⋊ C2×C4×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D45 |
| commutator, hypocenter | C5×C9 | C22×C4 |
| Frattini | C2×C3 | C22×D15 |
| Fitting | C2×C4×C5×C9 | C2 |
| socle | C22×C3×C5 | C2×D3 |
| prime radical | C22×D45 | C2 |
| cosocle | C2×C5×C9 | C23 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C2×C4 | |
162 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
192 conjugacy classes — view the full character table.
| maximal subgroup of | #13083733, #13083863, C2×#10623539, … |
| maximal normal subgroup of | #13083733, #13083863, C2×#10623539, … |
| maximal quotient of | C4×Dic90, C42×D45, #13084520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 176);