TITLE: Explicit formulas Belyi maps for a rational dessin d'enfants
QUESTION [6 upvotes]: What are the best references for finding explicit formulas for Belyi maps for rational dessin d'enfants? 
I am most interested in a formula for the Belyi map that corresponds to a specific rational dessin d'enfant: it is clean, with four black vertices, each of valency 3, and with ramification indices above the infinity of 6,3,2,and 1. (I believe that there is only one such dessin d'enfant).  Ideally, it would be great to have tables or software for other similar examples.

REPLY [2 votes]: You are right that there is only one such dessin. To see that,
type the following in Maple 2019:

with(GroupTheory):
  FindDessins([3,3,3,3],[2,2,2,2,2,2],[6,3,2,1]);

You can find more information on:
https://www.math.fsu.edu/~hoeij/Heun
In the main file there you can find:
   S3: ModuliField:    Q.
       Obstruction:    none
       Branch type:    0:[3^4]  1:[2^6]  infty:[1,2,3,6]
       Decomposition:  S47(deg 3) 1/S34(deg 4) 
       Galois group:   A4 x S3.  GAP transitive group ID = 12T43.  Order = 72

The name "S3: ..." refers to rational functions in the file:
https://www.math.fsu.edu/~hoeij/Heun/BelyiMaps
which gives this rational function:
$S_3 := {\frac {4\, \left( {x}^{3}-6\,x-6 \right) ^{3}{x}^{3}}{27\, \left( x+1
 \right) ^{3} \left( 2\,x+3 \right) ^{2} \left( x-3 \right) }}$