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Based on recent work of Spaltenstein [ 14 ] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple groups      3 D 4 ( q ) {}^3{D_4}(q)       is given. As an application we obtain a classification of the irreducible characters of      3 D 4 ( q ) {}^3{D_4}(q)       into      r r      -blocks for all primes      r &gt; 0 r &gt; 0      . This enables us to verify Brauer’s height zero conjecture, his conjecture on the bound of irreducible characters belonging to a give block, and the Alperin-McKay conjecture for the simple triality groups      3 D 4 ( q ) {}^3{D_4}(q)      . It also follows that for every prime      r r       there are blocks of defect zero in      3 D 4 ( q ) {}^3{D_4}(q)      .

Character table and blocks of finite simple triality groups ³𝐷₄(𝑞)