Klein's ten “dessins d'enfants” of degree $11$: theme with variations

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 the 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 $\mathrm{PSL}_n(q)$ arise as permutation groups and monodromy groups of degree $p$ (an open problem in group theory).