Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C5×C9×A4 |
| Fitting | C22×C5×C9 ⋊ C3 |
| derived, Sylow | C22 ⋊ C3×C5×C9 |
| Hall | C22×C5 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C9 | A4 |
| commutator, hypocenter | C22 | C3×C5×C9 |
| Frattini | C3 | C3×C5×A4 |
| Fitting | C22×C5×C9 | C3 |
| socle | C22×C3×C5 | C32 |
| prime radical | C3×C5×A4 | C3 |
| cosocle | C22×C3 | C32×C5 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C5×C9 | |
46 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 57);