Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C27 · Schur multiplier (H2) = C2 |
| direct product | C5×C22⋊C27 |
| unambiguous semidirect product | C22⋊(C5×C27) |
| unambiguous semidirect product | (C22×C5)⋊C27 |
| derived, Sylow | C22 ⋊ C5×C27 |
| Hall | C22×C5 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C9 | A4 |
| commutator, hypocenter | C22 | C5×C27 |
| Frattini | C9 | C5×A4 |
| Fitting | C22×C5×C9 | C3 |
| socle, prime radical | C22×C3×C5 | C9 |
| cosocle | C22×C9 | C3×C5 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C5×C9 | |
22 conjugacy classes of subgroups.
| transitive degree | μt(G) = 270 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 8);