  First, we prove the Kac-Wakimoto conjecture on modular invariance of
characters of exceptional affine W-algebras. In fact more generally we prove
modular invariance of characters of all lisse W-algebras obtained through
Hamiltonian reduction of admissible affine vertex algebras. Second, we prove
the rationality of a large subclass of these W-algebras, which includes all
exceptional W-algebras of type A and lisse subregular W-algebras in simply
laced types. Third, for the latter cases we compute S-matrices and fusion
rules. Our results provide the first examples of rational W-algebras associated
with non-principal distinguished nilpotent elements, and the corresponding
fusion rules are rather mysterious.
