Bounds on automorphism groups of surfaces of general type with negative c_2

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $S$ be a minimal smooth projective surface of general type over $k$. If the second Chern class satisfies $c_2(S)<0$, then the order of the automorphism group satisfies $\mathrm{\mid }{\mathrm{A}\mathrm{u}\mathrm{t}}_k(S)\mathrm{\mid }<2598(K_S^2{)}^4$, and the order of every abelian subgroup $G\subset {\mathrm{A}\mathrm{u}\mathrm{t}}_k(S)$ satisfies $\mathrm{\mid }G\mathrm{\mid }<81(K_S^2{)}^3$.