Finite Group DatabasebetaFinite Group Databasebeta| order | 6390 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C9⋊(D71⋊C5) |
| unambiguous semidirect product | (C71⋊C5)⋊D9 |
| unambiguous semidirect product | D639⋊C5 |
| Fitting, derived, Hall | C9×C71 ⋊ C2×C5 |
| Hall | C9×C71⋊C5 ⋊ C2 |
| Hall | D639 ⋊ C5 |
| Hall | C71⋊C5 ⋊ D9 |
| Sylow | C71 ⋊ C5×D9 |
| Sylow | C9 ⋊ D71⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C9×C71 | C2×C5 |
| Frattini | C3 | C3⋊(D71⋊C5) |
| socle | C3×C71 | C5×D3 |
| LCS | G ▷ C9×C71 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,5,71,3|b^3=e^2,b^a=b^2e,d^a=d^70,e^a=e^2,d^c=d^5493 conjugacy classes — view the full character table.
| maximal subgroup of | D9×D71⋊C5, C2×C9⋊(D71⋊C5), C5×C9⋊(D71⋊C5), … |
| maximal normal subgroup of | D9×D71⋊C5, C2×C9⋊(D71⋊C5), C5×C9⋊(D71⋊C5) |
| maximal quotient of | C9⋊(Dic71⋊C5), C2×C9⋊(D71⋊C5), C9⋊(D71⋊C25), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6390, 12);