Finite Group DatabasebetaFinite Group Databasebeta| order | 6996 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D33⋊D53 |
| unambiguous semidirect product | D159⋊D11 |
| unambiguous semidirect product | D583⋊D3 |
| Fitting, socle, derived, Hall | C3×C11×C53 ⋊ C22 |
| Hall | C11×C53 ⋊ C2×D3 |
| Hall | C3×C53 ⋊ C2×D11 |
| Sylow | C53 ⋊ D3×D11 |
| Hall | C3×C11 ⋊ C2×D53 |
| Sylow | C11 ⋊ D3×D53 |
| Sylow | C3 ⋊ D11×D53 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C11×C53 | C22 |
| LCS | G ▷ C3×C11×C53 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,11,53|d^a=d^10,e^a=e^52,c^b=c^2,d^b=d^10489 conjugacy classes — view the full character table.
| maximal subgroup of | D99⋊D53, C3×D33⋊D53, D159⋊D33, … |
| maximal normal subgroup of | C3×D33⋊D53, D159⋊AGL(1,𝔽11), C5×D33⋊D53 |
| maximal quotient of | D99⋊D53, C3×D33⋊D53, D159⋊D33, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6996, 20);