  In 1978, Stanley constructed an example of an Artinian Gorenstein (AG) ring
$A$ with non-unimodal $H$-vector $(1,13,12,13,1)$. Migliore-Zanello later
showed that for regularity $r=4$, Stanley's example has the smallest possible
codimension $c$ for an AG ring with non-unimodal $H$-vector. The weak Lefschetz
property (WLP) has been much studied for AG rings; it is easy to show that an
AG ring with non-unimodal $H$-vector fails to have WLP. In codimension $c=3$ it
is conjectured that all AG rings have WLP. For $c=4$, Gondim showed that WLP
always holds for $r \le 4$ and gives a family where WLP fails for any $r \ge
7$, building on an earlier example of Ikeda of failure of WLP for $r=5$. In
this note we study the minimal free resolution of $A$ and relation to Lefschetz
properties (both weak and strong) and Jordan type for $c=4$ and $r \le 6$.
