Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C27 · Schur multiplier (H2) = C1 |
| direct product | C5×D3×C27 |
| unambiguous semidirect product | C3⋊(C2×C5×C27) |
| Fitting, Hall | C3×C5×C27 ⋊ C2 |
| derived | C3 ⋊ C2×C5×C27 |
| Sylow | C3×C27 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C27 | D3 |
| commutator, hypocenter | C3 | C2×C5×C27 |
| Frattini | C9 | C3×C5×D3 |
| Fitting | C3×C5×C27 | C2 |
| socle | C32×C5 | C2×C9 |
| prime radical | C3×C5×D3 | C9 |
| cosocle | C3×C9 | C2×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C5×C27 | |
38 conjugacy classes of subgroups.
| transitive degree | μt(G) = 270 |
405 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D3×D5×C27, C5×D3×D27, D3×D135, … |
| maximal normal subgroup of | D3×D5×C27, C5×D3×D27, D3×D135, … |
| maximal quotient of | C5×Dic3×C27, C2×C5×D3×C27 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 19);