On p-parts of Brauer character degrees and p-regular conjugacy class sizes

Let $G$ be a finite group, $p$ a prime, and $IB{r}_p(G)$ the set of irreducible $p$-Brauer characters of $G$. Let ${\bar{e}}_p(G)$ be the largest integer such that $p^{{\bar{e}}_p(G)}$ divides $\chi (1)$ for some $\chi \in IB{r}_p(G)$. We show that $\mathrm{\mid }G:O_p(G){\mathrm{\mid }}_p\le p^{k{\bar{e}}_p(G)}$ for an explicitly given constant $k$. We also study the analogous problem for the $p$-parts of the conjugacy class sizes of $p$-regular elements of finite groups.