Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
| direct product | C23×C9×D5 |
| unambiguous semidirect product | C5⋊(C24×C9) |
| unambiguous semidirect product | (C2×C5)⋊(C23×C9) |
| Fitting | C23×C5×C9 ⋊ C2 |
| derived, Sylow | C5 ⋊ C24×C9 |
| Hall | C5×C9 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C9 | D5 |
| commutator, hypocenter | C5 | C24×C9 |
| Frattini | C3 | C23×C3×D5 |
| Fitting | C23×C5×C9 | C2 |
| socle | C23×C3×C5 | C2×C3 |
| prime radical | C23×C3×D5 | C3 |
| cosocle | C3×C5 | C24×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23×C9 | |
402 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#10623562, D5×#3676, #13084328, … |
| maximal normal subgroup of | C2×#10623562, D5×#3676, #13084328, … |
| maximal quotient of | C22×C9×Dic10, C22×C4×C9×D5, C22×C9×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 405);