  This paper and its sequel describe the irreducible representations of the
rational Cherednik algebra $H_c(W)$ for a finite Coxeter group $W$ of type
$H_4$, $F_4$ with equal parameters, $E_6$, $E_7$, and $E_8$, when $c$ is not a
half-integer. Herein appear the decomposition matrices of Category
$\mathcal{O}_c(W)$ for $W=H_4$, $E_6$, and $E_7$, as well as a classification
of finite-dimensional representations of $H_c(W)$ for these groups and for
$H_c(F_4)$ with equal parameters.
