  Superconformal indices (SCIs) of 4d ${\mathcal N}=4$ SYM theories with simple
gauge groups are described in terms of elliptic hypergeometric integrals. For
$F_4, E_6, E_7, E_8$ gauge groups this yields first examples of integrals of
such type. S-duality transformation for G_2 and F_4 SCIs is equivalent to a
change of integration variables. Equality of SCIs for SP(2N) and SO(2N+1) group
theories is proved in several important special cases. Reduction of SCIs to
partition functions of 3d $\mathcal{N}=2$ SYM theories with one matter field in
the adjoint representation is investigated, corresponding 3d dual partners are
found, and some new related hyperbolic beta integrals are conjectured.
