Let $p>2$ be prime. Then for abstract reasons the special linear group $\text{SL}_2({\mathbb F}_p)$ possesses a free action on some sphere (one has to check that any abelian subgroup of $\text{SL}_2({\mathbb F}_p)$ is cyclic and that there's at most one element of order $2$).

Does somebody know a concrete example for such a free action for general $p$? (For $p=5$, for example, $\text{SL}_2({\mathbb F}_p)$is the binary icosahedral group which is a subgroup of ${\mathbb S}^3$ thus acting freely on it by multiplication; I'd like to know if there's one single action that can be written down for all $p$ simultaneously).