A family of multiply extended grids

We construct an infinite family {Gamma(n)}(n=5) of finite connected graphs Gamma(n) that are multiple extensions of the well-known ''extended grid'' discovered in [1] (which is isomorphic to Gamma(5)). The graphs Gamma(n) are locally Gamma(n-1) for n > 5, and have the following property: the automorphism group G(n) of Gamma(n) permutes transitively the maximal cliques of Gamma(n) (which are n-cliques) and the stabilizer of some n-clique x of Gamma(n) in G(n) induces Sigma(n) on the vertices of pi. Furthermore we show that the clique complexes of the graphs Gamma(n) are simply connected.