Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C3⋊D15)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C36 |
| unambiguous semidirect product | C3⋊(C3⋊(C3⋊D15)) |
| unambiguous semidirect product | C5⋊(C3⋊(C3⋊(C3⋊D3))) |
| Fitting, socle, derived, Hall | C34×C5 ⋊ C2 |
| Sylow | C34 ⋊ D5 |
| Sylow | C5 ⋊ C3⋊(C3⋊(C3⋊D3)) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C34×C5 | C2 |
| LCS | G ▷ C34×C5 | |
| UCS | C1 | |
848 conjugacy classes of subgroups.
| transitive degree | μt(G) = 405 |
Polycyclic presentation:
2,3^4,5|b^a=b^2,c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^4204 conjugacy classes — view the full character table.
| maximal subgroup of | #13108536, D5×C3⋊(C3⋊(C3⋊D3)), D3×C3⋊(C3⋊D15), … |
| maximal normal subgroup of | #13108536, D5×C3⋊(C3⋊(C3⋊D3)), D3×C3⋊(C3⋊D15), … |
| maximal quotient of | C3⋊(C3⋊(C3⋊Dic15)), C2×C3⋊(C3⋊(C3⋊D15)) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 112);