We compute the conjugacy classes of elements and the character tables of the parabolic subgroups of Steinberg's triality groups 3 D 4 (q), where q is a power of an odd prime. Our main tools are Clifford theory applied to the Levi decomposition of the parabolic subgroups and the decomposition of restrictions of the unipotent characters of 3 D 4 (q). For many of the calculations we use the CHEVIE package.

We compute the conjugacy classes and character table of a Borel subgroup of the Ree groups 2 F 4 (2 2n+1 ) for all n 1 and prove that these Borel subgroups are M -groups. We determine the degrees of the irreducible characters of the Sylow-2-subgroups of 2 F 4 (2 2n+1 ) and show that the IsaacsMalle-Navarro version of the McKay conjecture holds for 2 F 4 (2 2n+1 ) in characteristic 2. For most of the calculations we use CHEVIE.

We compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of the simple Ree groups 2 F 4(q 2 ). For one of the maximal parabolic subgroups, we find an irreducible character of the unipotent radical that does not extend to its inertia subgroup.

We complete the ℓ-modular decomposition numbers of the unipotent characters in the principal block of the special orthogonal groups SO 7 (q) and the symplectic groups Sp 6 (q) for all prime powers q and all odd primes ℓ different from the defining characteristic.

This paper is a contribution to the general program introduced by Isaacs, Malle and Navarro to prove the McKay conjecture in the representation theory of finite groups. We develop new methods for dealing with simple groups of Lie type in the defining characteristic case. Using a general argument based on the representation theory of connected reductive groups with disconnected center, we show that the inductive McKay condition holds if the Schur multiplier of the simple group has order 2. As a consequence, the simple groups PΩ 2m+1 (p n ) and PSp 2m (p n ) are "good" for p > 2 and the simple groups E 7 (p n ) are "good" for p > 3 in the sense of Isaacs, Malle and Navarro. We also describe the action of the diagonal and field automorphisms on the semisimple and the regular characters.

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