A new family of locally $5$-arc transitive graphs of pushing up type with respect to the prime 3

Let q be a power of the prime 3. A locally 5-arc transitive G-graph of pushing up type is constructed for each value of q. For $q=3$, the G-graph constructed provides an example of a graph with a vertex stabilizer amalgam of shape E₁ in the sense of [1]. Whereas, for the other values of q, the vertex stabilizer amalgam of the G-graph is of a previously unknown shape. In particular, for q ≠ 3, these graphs are the first examples of locally 5-arc transitive graphs containing a vertex z for which the group that fixes all 3-arcs originating at z is non-trivial.