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We construct the graphs of Hoffman-Singleton, Higman-Sims, and McLaughlinfrom certain relations on the set of non-singular conics totally tangent to theHermitian curve of degree 6 in characteristic 5. We then interpret thisgeometric construction in terms of the subgroup structure of the automorphismgroup of this Hermitian curve.

The graphs of Hoffman-Singleton, Higman-Sims, and McLaughlin, and the Hermitian curve of degree 6 in characteristic 5