Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C5 · Schur multiplier (H2) = C33 |
| direct product | C33×C5×D3 |
| unambiguous semidirect product | C3⋊(C2×C33×C5) |
| Fitting, socle, Hall | C34×C5 ⋊ C2 |
| derived | C3 ⋊ C2×C33×C5 |
| Sylow | C34 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C33×C5 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C33×C5 |
| Frattini | C1 | G |
| Fitting, socle | C34×C5 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C33×C5 | |
380 conjugacy classes of subgroups.
| transitive degree | μt(G) = 270 |
405 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C33×D3×D5, C32×C5×D32, C32×D3×D15, … |
| maximal normal subgroup of | C33×D3×D5, C32×C5×D32, C32×D3×D15, … |
| maximal quotient of | C33×C5×Dic3, C2×C33×C5×D3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 105);