Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of C3⋊D135 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊D135 |
| unambiguous semidirect product | C5⋊(C3⋊D27) |
| unambiguous semidirect product | C27⋊D15 |
| Fitting, derived, Hall | C3×C5×C27 ⋊ C2 |
| Sylow | C3×C27 ⋊ D5 |
| Sylow | C5 ⋊ C3⋊D27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C5×C27 | C2 |
| Frattini | C9 | C3⋊D15 |
| socle | C32×C5 | D9 |
| LCS | G ▷ C3×C5×C27 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 405 |
Polycyclic presentation:
2,3^2,5,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^2,d^a=d^4,e^a=e^2f^2,f^a=f^2204 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C3⋊D27, D3×D135, D15×D27, … |
| maximal normal subgroup of | D5×C3⋊D27, D3×D135, D15×D27, … |
| maximal quotient of | C3⋊Dic135, C2×C3⋊D135 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 49);