  Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known
that the degrees of the irreducible characters of the Sylow $p$-subgroup of
$GL_n(\F_q)$ are powers of $q$ by Issacs. On the other hand Sangroniz showed
that this is true for a Sylow $p$-subgroup of a classical group defined over
$\F_q$ if and only if $p$ is odd. For the classical groups of Lie type $B$, $C$
and $D$ the only bad prime is 2. For the exceptional groups there are others.
In this paper we construct irreducible characters for the Sylow $p$-subgroups
of the Chevalley groups $D_4(q)$ with $q=2^f$ of degree $q^3/2$. Then we use an
analogous construction for $E_6(q)$ with $q=3^f$ to obtain characters of degree
$q^7/3$, and for $E_8(q)$ with $q=5^f$ to obtain characters of degree
$q^{16}/5.$ This helps to explain why the primes 2, 3 and 5 are bad for the
Chevalley groups of type $E$ in terms of the representation theory of the Sylow
$p$-subgroup.
