Finite Group DatabasebetaFinite Group Databasebeta| order | 7686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| Fitting, socle, Hall | C2×C7×C61 ⋊ C32 |
| derived, Hall | C7×C61 ⋊ C2×C32 |
| Hall | C2×C61 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C61 ⋊ C2×C3×ΓL(1,𝔽8) |
| Hall | C2×C7 ⋊ C3×C61⋊C3 |
| Sylow | C7 ⋊ C2×C3×C61⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×C61⋊C3 |
| commutator, hypocenter | C7×C61 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C61 | C32 |
| LCS | G ▷ C7×C61 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7686, 29);