Finite Group DatabasebetaFinite Group Databasebeta| order | 7686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| Fitting, socle, derived, Hall | C7×C61 ⋊ C2×C32 |
| Hall | ΓL(1,𝔽8)×C61⋊C3 ⋊ C2 |
| Hall | D7×C61 ⋊ C32 |
| Sylow | C61 ⋊ C3×AGL(1,𝔽7) |
| Hall | D7 ⋊ C3×C61⋊C3 |
| Sylow | C7 ⋊ C2×C3×C61⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C61 | C2×C32 |
| LCS | G ▷ C7×C61 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7686, 27);