Recognition by spectrum for finite simple groups with orders having prime divisors at most 17

The spectrum $\omega(G)$ of a group $G$ is the set of its element orders. We write $h(G)$ to denote the number of pairwise non-isomorphic finite groups $H$ with $\omega(H)=\omega(G)$. We say that $G$ is recognizable by spectrum if $h(G)=1$ and that $G$ is a group with solved recognition-by-spectrum problem if $h(G)$ is known. In the paper we prove that the groups $C_3(4)$ and $D_4(4)$ are recognizable by spectrum. It follows from this result that the recognition-by-spectrum problem is solved for all finite simple groups with orders having prime divisors at most $17$.