Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D45 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D45 |
| unambiguous semidirect product | (C2×C5)⋊(C3×D9) |
| Fitting | C2×C3×C5×C9 ⋊ C2 |
| derived | C5×C9 ⋊ C22×C3 |
| Hall | C3×C5×C9 ⋊ C22 |
| Sylow | C3×C9 ⋊ C2×D5 |
| Sylow | C5 ⋊ C2×C3×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D45 |
| commutator, cosocle, hypocenter | C5×C9 | C22×C3 |
| Frattini | C3 | C2×C3×D15 |
| Fitting | C2×C3×C5×C9 | C2 |
| socle | C2×C32×C5 | D3 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C2×C3 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#67079, C3×#67080, C3×#67083, … |
| maximal normal subgroup of | C3×#67079, C3×#67080, C3×#67083, … |
| maximal quotient of | C3×Dic90, C3×C4×D45, C3×D180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 69);