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In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H -vector (1 , 13 , 12 , 13 , 1). Migliore-Zanello show in (Migliore and Zanello, 2017) 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 shows in (Gondim, 2017) that WLP always holds for r ≤ 4 and gives a family where WLP fails for any r ≥ 7 , building on Ikeda's example (Ikeda, 1996) of failure 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 ≤ 6. [ABSTRACT FROM AUTHOR]