Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C9 · Schur multiplier (H2) = C22 |
| direct product | C22×C5×C22⋊C9 |
| derived | C22 ⋊ C22×C5×C9 |
| Hall | C24×C5 ⋊ C9 |
| Sylow | C24 ⋊ C5×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C5 | A4 |
| commutator, hypocenter | C22 | C22×C5×C9 |
| Frattini | C3 | C22×C5×A4 |
| Fitting, socle, prime radical | C24×C3×C5 | C3 |
| cosocle | C22×C3 | C22×C3×C5 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C22×C3×C5 | |
136 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 403);