Vertex-transitive non-Cayley graphs with arbitrarily large vertex-stabilizer

A construction is given for an infinite family {Gamma(n)} of finite vertex-transitive non-Cayley graphs of fixed valency with the property that the order of the vertex-stabilizer in the smallest vertex-transitive group of automorphisms of Gamma(n) is a strictly increasing function of n. For each n the graph is 4-valent and are-transitive, with automorphism group a symmetric group of large prime degree p > 2(2n+2). The construction uses Sierpinski's gasket to produce generating permutations for the vertex-stabilizer (a large 2-group).