We study the free resolutions of some Artin Gorenstein algebras of Hilbert function
(
1
,
4
,
k
,
k
,
4
,
1
)
(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
k=4
we prove that the Hilbert function alone fixes the Betti table. For higher
k
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.