Finite Group DatabasebetaFinite Group Databasebeta| order | 7686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×ΓL(1,𝔽8)⋊D61 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3×D61) |
| Fitting, socle | C3×C7×C61 ⋊ C2×C3 |
| derived, Hall | C7×C61 ⋊ C2×C32 |
| Hall | C3×ΓL(1,𝔽8)×C61 ⋊ C2 |
| Hall | D427 ⋊ C32 |
| Hall | C3×ΓL(1,𝔽8) ⋊ D61 |
| Sylow | C61 ⋊ C3×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C32×D61 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽8)⋊D61 |
| commutator, hypocenter | C7×C61 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7×C61 | C2×C3 |
| LCS | G ▷ C7×C61 | |
| UCS | C1 ◁ C3 | |
48 conjugacy classes of subgroups.
471 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7686, 36);