Any finite group is the group of symmetries of some map (atom-bifurcation) / *E. A. Kudryavtseva, A. T. Fomenko.* // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2013. № 3. P. 21-29
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 0].

Maps are studied, i.e. cell decompositions of closed two-dimensional surfaces, or two-dimensional atoms, which encode bifurcations of Liouville fibrations of nondegenerate integrable Hamiltonian systems. Any finite group *G* is proved to be the symmetry group of an orientable map (of an atom). Moreover one such a map *X*(*G*) is constructed algorithmically. Upper bounds are obtained for the minimal genus M*g*(*G*) of an orientable map with the given symmetry group *G*, and for the minimal number of vertices, edges and sides of such maps.

*Key words*:
finite group, orientable map, symmetry group of a map, group action on a closed surface.