  At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent
group-theoretical structure for Ramanujan's mock theta functions analogous to
Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can
be written in terms of $R(\zeta_p,q)$, where $R(z,q)$ is the two-variable
generating function of Dyson's rank function and $\zeta_p$ is a primitive
$p$-th root of unity. In his lost notebook Ramanujan gives the $5$-dissection
of $R(\zeta_5,q)$. This result is related to Dyson's famous rank conjecture
which was proved by Atkin and Swinnerton-Dyer. In 2016 the first author showed
that there is an analogous result for the $p$-dissection of $R(\zeta_p,q)$ when
$p$ is any prime greater than $3$, by extending work of Bringmann and Ono, and
Ahlgren and Treneer. It was also shown how the group $\Gamma_1(p)$ acts on the
elements of the $p$-dissection of $R(\zeta_p,q)$. We extend this to the group
$\Gamma_0(p)$, thus revealing new and surprising symmetries for Dyson's rank
function.
