  We reinterpret ideas in Klein's paper on transformations of degree~$11$ from
the modern point of view of dessins d'enfants, and extend his results by
considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is
prime. In many cases we determine the passports and monodromy groups of these
dessins, and in a few small cases we give drawings which are topologically (or,
in certain examples, even geometrically) correct. We use the Bateman--Horn
Conjecture and extensive computer searches to support a conjecture that there
are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for some prime power
$q$, in which case infinitely many groups ${\rm PSL}_n(q)$ arise as permutation
groups and monodromy groups of degree $p$ (an open problem in group theory).
