Codimension two Artinian algebras over a field of characteristic zero have the strong Lefschetz property. It is open whether Artinian Gorenstein algebras of codimension three satisfy the weak or strong Lefschetz properties. Harima, Migliore, Nagel, and Watanabe proved that complete intersection algebras of codimension three satisfy the weak Lefschetz property. We extend this result and show the presence of the strong Lefschetz property for Artinian Gorenstein algebras with low sperner numbers; i.e. maximum value of the Hilbert function. When the characteristic is sufficiently large, we show that these algebras are almost strong Lefschetz. This talk is based on a joint work with Abdallah, Iarrobino, Seceleanu, and Yaméogo.