  Codimension two Artinian algebras $A$ have the strong and weak Lefschetz
properties provided the characteristic is zero or greater than the socle
degree. It is open to what extent such results might extend to codimension
three AG algebras - the most promising results so far have concerned the weak
Lefschetz property for such algebras. We here show that every standard-graded
codimension three Artinian Gorenstein algebra $A$ having low maximum value of
the Hilbert function - at most six - has the strong Lefschetz property,
provided that the characteristic is zero. When the characteristic is greater
than the socle degree of $A$, we show that $A$ is almost strong Lefschetz. This
quite modest result is nevertheless arguably the most encompassing so far
concerning the strong Lefschetz property for graded codimension three AG
algebras.
