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Let   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(  q  ) are powers of  q . On the other hand Sangroniz (2003) showed that this is true for a Sylow  p -subgroup of a classical group defined over   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.

On the character degrees of Sylow p -subgroups of Chevalley groups G ( p f ) of type E