Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C5 · Schur multiplier (H2) = C26 |
| direct product | C23×C3×C5×D3 |
| unambiguous semidirect product | C3⋊(C24×C3×C5) |
| unambiguous semidirect product | (C2×C3)⋊(C23×C3×C5) |
| Fitting, socle | C23×C32×C5 ⋊ C2 |
| derived | C3 ⋊ C24×C3×C5 |
| Hall | C32×C5 ⋊ C24 |
| Sylow | C32 ⋊ C24×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3×C5 | D3 |
| commutator, cosocle, hypocenter | C3 | C24×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32×C5 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C3×C5 | |
568 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 836);