Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×C9×Dic3 |
| unambiguous semidirect product | C3⋊(C4×C5×C9) |
| derived | C3 ⋊ C4×C5×C9 |
| Hall | C3×C5×C9 ⋊ C4 |
| Sylow | C3×C9 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5×C9 | D3 |
| commutator, hypocenter | C3 | C4×C5×C9 |
| Frattini | C2×C3 | C3×C5×D3 |
| Fitting | C2×C3×C5×C9 | C2 |
| socle, prime radical | C2×C32×C5 | C2×C3 |
| cosocle | C2×C32 | C2×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C5×C9 | |
44 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C9×D5×Dic3, C9×#1200, C9×#1202, … |
| maximal normal subgroup of | C9×D5×Dic3, C9×#1200, C9×#1202, … |
| maximal quotient of | C5×C9×C3⋊C8, C2×C5×C9×Dic3, C5×C9×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 17);