The irreducible characters of the Sylow $p$-subgroups of the Chevalley groups ${\mathrm{D}}_6(p^f)$ and ${\mathrm{E}}_6(p^f)$

We parametrize the set of irreducible characters of the Sylow $p$-subgroups of the Chevalley groups ${\mathrm{D}}_6(q)$ and ${\mathrm{E}}_6(q)$, for an arbitrary power $q$ of any prime $p$. In particular, we establish that the parametrization is uniform for $p\ge 3$ in type ${\mathrm{D}}_6$ and for $p \ge 5$intype$$\mathrm{E}$_6$,whiletheprime$2$intype$$\mathrm{D}$_6$ and the primes $2,$ $3$ in type ${\mathrm{E}}_6$ yield character degrees of the form $q^m\mathrm{/}{p}^i$ which force a departure from the generic situations. Also for the first time in our analysis we see a family of irreducible characters of a classical group of degree $q^m\mathrm{/}{p}^i$ where $i>1$ which occurs in type ${\mathrm{D}}_6$.