Finite Group DatabasebetaFinite Group Databasebeta| order | 7686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | C2×#16838636 |
| derived, Hall | C7×C61 ⋊ C2×C9 |
| Hall | C2×C7×C61 ⋊ C9 |
| Hall | C2×C61 ⋊ C7⋊C9 |
| Sylow | C61 ⋊ C2×C7⋊C9 |
| Hall | C2×C7 ⋊ C61⋊C9 |
| Sylow | C7 ⋊ C2×C61⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #13046197 |
| commutator, hypocenter | C7×C61 | C2×C9 |
| Frattini | C3 | C2×#13046197 |
| Fitting, socle | C2×C3×C7×C61 | C3 |
| LCS | G ▷ C7×C61 | |
| UCS | C1 ◁ C2×C3 | |
24 conjugacy classes of subgroups.
870 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#16838636, #29506705, #29506709, … |
| maximal normal subgroup of | C4×#16838636, #29506705, #29506709, … |
| maximal quotient of | C4×#16838636, C22×#16838636, C2×C5×#16838636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7686, 4);