Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C5 · Schur multiplier (H2) = C24×C3 |
| direct product | C3×C5×C22⋊A4 |
| unambiguous semidirect product | (C22×C3)⋊(C5×A4) |
| unambiguous semidirect product | (C22×C3×C5)⋊A4 |
| Fitting, socle | C24×C3×C5 ⋊ C3 |
| derived, Sylow | C24 ⋊ C32×C5 |
| Hall | C24×C5 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | C22⋊A4 |
| commutator, cosocle, hypocenter | C24 | C32×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C3×C5 | |
164 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, 834);