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In this paper we study the geometry of the $14$ families of K3 surfaces ofPicard number four with finite automorphism group, whose N\'eron-Severilattices have been classified by \`E.B. Vinberg. We provide projective models,we identify the degrees of a generating set of the Cox ring and in some caseswe prove the unirationality of the associated moduli space.

Mori dream K3 surfaces of Picard number four: projective models and Cox rings