Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C9 · Schur multiplier (H2) = C24 |
| direct product | C5×C22⋊(C22⋊C9) |
| unambiguous semidirect product | C22⋊(C5×C22⋊C9) |
| unambiguous semidirect product | (C22×C5)⋊(C22⋊C9) |
| derived, Sylow | C24 ⋊ C5×C9 |
| Hall | C24×C5 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | C22⋊A4 |
| commutator, hypocenter | C24 | C5×C9 |
| Frattini | C3 | C5×C22⋊A4 |
| Fitting, socle, prime radical | C24×C3×C5 | C3 |
| cosocle | C24×C3 | C3×C5 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C3×C5 | |
122 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 404);