Which Faber–Moore–Chen digraphs are Cayley digraphs?

The problem of finding the largest graphs and digraphs of given degree and diameter is known as the ‘degree–diameter’ problem. One of the families of largest known vertex-transitive digraphs of given degree and diameter is the Faber–Moore–Chen digraphs. In our contribution we will classify those Faber–Moore–Chen digraphs that are Cayley digraphs.