Finite Group DatabasebetaFinite Group Databasebeta| order | 7686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C32×D427 |
| Fitting, socle, Hall | C32×C7×C61 ⋊ C2 |
| derived, Hall | C7×C61 ⋊ C2×C32 |
| Hall | C32×C61 ⋊ D7 |
| Hall | C32×C7 ⋊ D61 |
| Sylow | C61 ⋊ C32×D7 |
| Sylow | C7 ⋊ C32×D61 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | D427 |
| commutator, hypocenter | C7×C61 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7×C61 | C2 |
| LCS | G ▷ C7×C61 | |
| UCS | C1 ◁ C32 | |
48 conjugacy classes of subgroups.
1935 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×D7×D61, C3×D3×D427, C3×D21⋊D61, … |
| maximal normal subgroup of | C32×D7×D61, C3×D3×D427, C3×D21⋊D61, … |
| maximal quotient of | C32×Dic427, C2×C32×D427, C32×C5×D427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7686, 64);