Finite Group DatabasebetaFinite Group Databasebeta| order | 5412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D33⋊D41 |
| unambiguous semidirect product | D123⋊D11 |
| unambiguous semidirect product | D451⋊D3 |
| Fitting, socle, derived, Hall | C3×C11×C41 ⋊ C22 |
| Hall | C11×C41 ⋊ C2×D3 |
| Hall | C3×C41 ⋊ C2×D11 |
| Sylow | C41 ⋊ D3×D11 |
| Hall | C3×C11 ⋊ C2×D41 |
| Sylow | C11 ⋊ D3×D41 |
| Sylow | C3 ⋊ D11×D41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C11×C41 | C22 |
| LCS | G ▷ C3×C11×C41 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,11,41|d^a=d^10,e^a=e^40,c^b=c^2,d^b=d^10381 conjugacy classes — view the full character table.
| maximal subgroup of | D99⋊D41, C3×D33⋊D41, D451⋊(C3⋊D3), … |
| maximal normal subgroup of | C3×D33⋊D41, #17985247⋊D3, #17985250⋊D3, … |
| maximal quotient of | D99⋊D41, C3×D33⋊D41, D451⋊(C3⋊D3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5412, 20);