The computation of Belyi pairs of 6-edged dessins d'enfants of genus 3 with automorphism groups of order 12 and 3

In this article, we study dessins d'enfants of genus three with six edges and only one vertex, having group of automorphisms of order at least three. We find all such dessins and explain the method of the enumeration. For each of them the Belyi pair is computed.