Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of D27⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×D27⋊C3 |
| unambiguous semidirect product | D27⋊(C3×C5) |
| unambiguous semidirect product | (C27⋊C3)⋊(C2×C5) |
| unambiguous semidirect product | (C5×D27)⋊C3 |
| Fitting, Hall | C5×C27⋊C3 ⋊ C2 |
| derived | C27 ⋊ C2×C3×C5 |
| Sylow | C27⋊C3 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D27⋊C3 |
| commutator, cosocle, hypocenter | C27 | C2×C3×C5 |
| Frattini | C9 | C3×C5×D3 |
| Fitting | C5×C27⋊C3 | C2 |
| socle | C3×C5 | C3×D9 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C5 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 135 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D27⋊C3, C2×C5×D27⋊C3, C5×AGL(1,ℤ/27) |
| maximal normal subgroup of | D5×D27⋊C3, C2×C5×D27⋊C3, C5×AGL(1,ℤ/27) |
| maximal quotient of | C5×(C27⋊C3)⋊C4, C2×C5×D27⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 20);