  We study the free resolutions of some Artin Gorenstein algebras of Hilbert
function $(1,4,k,k,4,1)$ and we prove that all such algebras have the Strong
Lefschetz property if they have the Weak Lefschetz property. In the case $k=4$
we prove that the Hilbert function alone fixes the betti table. For higher $k$
stronger conditions on the algebras are needed to fix the betti table. In
particular, if the algebra is a complete intersection or if it is defined by an
equigenerated ideal then the betti table is unique.
