Finite Group DatabasebetaFinite Group Databasebeta| order | 6996 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C53 · Schur multiplier (H2) = C2 |
| direct product | D3×D11×C53 |
| unambiguous semidirect product | (D3×C11)⋊(C2×C53) |
| unambiguous semidirect product | (C3×D11)⋊(C2×C53) |
| unambiguous semidirect product | D33⋊(C2×C53) |
| Fitting, socle, Hall | C3×C11×C53 ⋊ C22 |
| derived, Hall | C3×C11 ⋊ C22×C53 |
| Hall | C11×C53 ⋊ C2×D3 |
| Hall | C3×C53 ⋊ C2×D11 |
| Sylow | C11 ⋊ C2×D3×C53 |
| Sylow | C3 ⋊ C2×D11×C53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C53 | D3×D11 |
| commutator, hypocenter | C3×C11 | C22×C53 |
| Frattini | C1 | G |
| Fitting, socle | C3×C11×C53 | C22 |
| LCS | G ▷ C3×C11 | |
| UCS | C1 ◁ C53 | |
40 conjugacy classes of subgroups.
1113 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D9×D11×C53, C3×D3×D11×C53, D3×C53×D33, … |
| maximal normal subgroup of | C3×D3×D11×C53, D3×C53×AGL(1,𝔽11), C5×D3×D11×C53 |
| maximal quotient of | D9×D11×C53, C3×D3×D11×C53, D3×C53×D33, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6996, 17);