The existence of directed strongly regular graphs with parameters $(63,11,8,1,2)$

In this paper, we prove the existence of directed strongly regular graphs with parameters $(63,11,8,1,2)$. We show that there are exactly two non-isomorphic directed strongly regular graphs with parameters $(63,11,8,1,2)$ on which $PSL(2,8)$ acts as and automorphism group, transitively on the vertices of the digraphs. Both directed strongly regular graphs have PSL(2,8):Z₃ as the full automorphism group.