Synchronising primitive groups of diagonal type exist

Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that PSL(2,q)xPSL(2,q)$\operatorname{{\mathrm{PSL}}}(2,q)\times \operatorname{{\mathrm{PSL}}}(2,q)$ acting in its diagonal action on PSL(2,q)$\operatorname{{\mathrm{PSL}}}(2,q)$ is separating, and hence synchronising, for q=13$q=13$ and q=17$q=17$. Furthermore, we show that such groups are non-spreading for all prime powers q$q$.