A construction is given for an infinite family ${Gamma_n}$ of the 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 arc-transitive, with automorphism grou a symmetric group of large prime degree $p>2^{{n+2}}$. The construction uses Sierpinski's gasket to produce generating permutations fo the vertex-stablilizer (a large 2-group). |

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2/4/97

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