Finite Group DatabasebetaFinite Group Databasebeta| order | 7686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊((C61⋊C3)⋊D7) |
| unambiguous semidirect product | (C61⋊C3)⋊D21 |
| Fitting, socle, derived | C3×C7×C61 ⋊ C2×C3 |
| Hall | C3×C7×C61⋊C3 ⋊ C2 |
| Hall | C3×C61⋊C3 ⋊ D7 |
| Hall | C7×C61 ⋊ C3×D3 |
| Sylow | C61 ⋊ C3×D21 |
| Sylow | C7 ⋊ C3⋊(D61⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C7×C61 | C2×C3 |
| LCS | G ▷ C3×C7×C61 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,7,61|c^a=c^2,d^a=d^6,e^a=e^60,e^b=e^13246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7686, 56);