Vertex-transitive maps with Schläfli type {3,7}{3,7}

Among all equivelar vertex-transitive maps on a given closed surface SS, the automorphism groups of maps with Schläfli types {3,7}{3,7} and {7,3}{7,3} allow the highest possible order. We describe a procedure to transform all such maps into 11- or 22-orbit maps, whose symmetry type has been previously studied. In doing so we provide two procedures to find vertex-transitive maps with Schläfli type {3,7}{3,7} which are neither regular nor chiral. One of them is by means of operations while the other consists of interpreting them as generalized Cayley maps. We determine all such maps on surfaces with Euler characteristic −1≥χ≥−40−1≥χ≥−40.