Finite Group DatabasebetaFinite Group Databasebeta| order | 7692 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D641 |
| unambiguous semidirect product | C641⋊(C22×C3) |
| unambiguous semidirect product | (C2×C641)⋊(C2×C3) |
| Fitting, socle | C2×C3×C641 ⋊ C2 |
| derived, Sylow | C641 ⋊ C22×C3 |
| Hall | C3×C641 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D641 |
| commutator, hypocenter | C641 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C641 | C2 |
| LCS | G ▷ C641 | |
| UCS | C1 ◁ C2×C3 | |
20 conjugacy classes of subgroups.
1932 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C9×D641, C3×D3×D641, C2×C32×D641, … |
| maximal normal subgroup of | C2×C9×D641, C2×C32×D641, C2×C3×D641⋊C5, … |
| maximal quotient of | C2×C9×D641, C3×D3×D641, C2×C32×D641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7692, 12);