Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D3×D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | D3×C9×D5 |
| unambiguous semidirect product | (C5×D3)⋊(C2×C9) |
| unambiguous semidirect product | (C3×D5)⋊(C2×C9) |
| unambiguous semidirect product | D15⋊(C2×C9) |
| Fitting, Hall | C3×C5×C9 ⋊ C22 |
| derived | C3×C5 ⋊ C22×C9 |
| Sylow | C3×C9 ⋊ C2×D5 |
| Sylow | C5 ⋊ C2×D3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | D3×D5 |
| commutator, hypocenter | C3×C5 | C22×C9 |
| Frattini | C3 | C3×D3×D5 |
| Fitting | C3×C5×C9 | C22 |
| socle, cosocle | C32×C5 | C22×C3 |
| prime radical | C3×D3×D5 | C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C9 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C9×AGL(1,𝔽5), D3×C9⋊AGL(1,𝔽5), D3×D5×D9, … |
| maximal normal subgroup of | D3×C9×AGL(1,𝔽5), D3×C9⋊AGL(1,𝔽5), D3×D5×D9, … |
| maximal quotient of | C9×D5×Dic3, D3×C9×Dic5, C9×#1200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 45);