Cliques in Paley graphs of square order and in Peisert graphs

We study maximal cliques in the collinearity graphs of Desarguesian nets, give some structural results and some numerical information. In particular, we show for Desarguesian nets that the set consisting of a point $x$ together with all its neighbors on a line $L$ (with $x$ not on $L$) is contained in a unique maximal clique $C_{x,L}$ and determine the sizes and automorphism groups of such maximal cliques $C_{x,L}$ in all cases.