Synchronising primitive groups of diagonal type exist

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