Let $j(\tau)$ be the modular invariant. Let $\mathcal M_n^*$ be the set of all $2$-by-$2$ matrices with relatively prime integer entries and with determinant $n$. The modular equation is defined by 
$$\Phi_n(X,j(\tau))=\prod_{M\in \operatorname{SL}_2(\mathbf Z)\backslash\mathcal M_n^*}(X-j(M\tau)).$$
What is the Galois group of the polynomial $\Phi_n(X,X)$?

For more details on the modular equation, see

[Zagier: Elliptic Modular Forms and Their Applications, p. 68][1]

[Andrew Sutherland: The modular equation][2]


  [1]: https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf
  [2]: https://math.mit.edu/classes/18.783/2015/LectureNotes20.pdf