Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C1 |
| direct product | D5×C27⋊C3 |
| unambiguous semidirect product | C5⋊(C2×C27⋊C3) |
| unambiguous semidirect product | (D5×C27)⋊C3 |
| Fitting, Hall | C5×C27⋊C3 ⋊ C2 |
| Sylow | C5 ⋊ C2×C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C9 | C32×D5 |
| commutator, socle | C3×C5 | C2×C3×C9 |
| Fitting | C5×C27⋊C3 | C2 |
| prime radical | C32×D5 | C9 |
| cosocle | C5×C9 | C2×C32 |
| hypercenter | C27⋊C3 | D5 |
| hypocenter | C5 | C2×C27⋊C3 |
| LCS | G ▷ C3×C5 ▷ C5 | |
| UCS | C1 ◁ C9 ◁ C27⋊C3 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 135 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×C27⋊C3, (C27⋊C3)⋊AGL(1,𝔽5), D5×D27⋊C3, … |
| maximal normal subgroup of | AGL(1,𝔽5)×C27⋊C3, (C27⋊C3)⋊AGL(1,𝔽5), D5×D27⋊C3, … |
| maximal quotient of | Dic5×C27⋊C3, C2×D5×C27⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 9);