  Among all equivelar vertex-transitive maps on a given closed surface S, the
automorphism groups of maps with Schl\"afli types {3, 7} and {7, 3} allow the
highest possible order. We describe a procedure to transform all such maps into
1- or 2-orbit maps, whose symmetry type has been previously studied. In so
doing we provide a procedure to determine all vertex-transitive maps with
Schl\"afli type {3, 7} which are neither regular or chiral. We determine all
such maps on surfaces with Euler characteristic -1 \geq \c{hi} \geq -40.
