Primitive permutation groups with a suborbit of length 5 and   vertex-primitive graphs of valency 5
  We classify finite primitive permutation groups having a suborbit of length 5. As a corollary, we obtain a classification of finite vertex-primitive graphs of valency 5. In the process, we also classify finite almost simple groups that have a maximal subgroup isomorphic to $\mathrm{Alt}(5)$ or $\mathrm{Sym}(5)$. 
Joanna B. Fawcett, Michael Giudici, Cai Heng Li, Cheryl E. Praeger,   Gordon Royle, Gabriel Verret
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