Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of C5⋊(D27⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊(D27⋊C3) |
| unambiguous semidirect product | (C27⋊C3)⋊D5 |
| unambiguous semidirect product | D135⋊C3 |
| Fitting, Hall | C5×C27⋊C3 ⋊ C2 |
| derived | C5×C27 ⋊ C2×C3 |
| Sylow | C27⋊C3 ⋊ D5 |
| Sylow | C5 ⋊ D27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C5×C27 | C2×C3 |
| Frattini | C9 | C3×D15 |
| Fitting | C5×C27⋊C3 | C2 |
| socle | C3×C5 | C3×D9 |
| LCS | G ▷ C5×C27 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 135 |
Polycyclic presentation:
2,3^2,5,3^2|c^3=e,e^3=f,c^a=c^2e^2f^2,d^a=d^4,e^a=e^2f^2,f^a=f^2,c^b=cf87 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D27⋊C3, C2×C5⋊(D27⋊C3), #72247130 |
| maximal normal subgroup of | D5×D27⋊C3, C2×C5⋊(D27⋊C3), #72247130 |
| maximal quotient of | (C27⋊C3)⋊Dic5, C2×C5⋊(D27⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 33);