Etayo Gordejuela, J. JavierMartínez García, Ernesto2023-06-202023-06-202008N. L. Alling and N. Greenleaf. Foundations of the theory of Klein surfaces, Lecture Notes in Mathematics, vol. 219 (Springer, 1971). E. Bujalance. Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary. Pac. J. Math. 109 (1983), 279–289. M. D. E. Conder and P. Dobcs´anyi. Determination of all regular maps of small genus. J. Combin. Theory B81 (2001), 224–242. J. J. Etayo. Sobre grupos de automorfismos de superficies de Klein. Doctoral thesis, Universidad Complutense, Madrid (1983). J. J. Etayo and E. Mart´ınez. The real genus of cyclic by dihedral and dihedral by dihedral groups. J. Alg. 296 (2006), 145–156. G. Gromadzki. Abelian groups of automorphisms of compact non-orientable Klein surfaces without boundary. Comment. Math. Prace Mat. 28 (1989), 197–217. A. M. Macbeath. The classification of non-Euclidean crystallographic groups. Can. J. Math. 19 (1967), 1192–1205. C. L. May. The symmetric cross-cap number of a group. Glasgow Math. J. 41 (2001), 399–410. C. L. May and J. Zimmerman. There is a group of every strong symmetric genus. Bull. Lond. Math. Soc. 35 (2003), 433–439. R. Preston. Projective structures and fundamental domains on compact Klein surfaces. Thesis, University of Texas (1975). D. Singerman. Automorphisms of compact non-orientable Riemann surfaces. Glasgow Math. J. 12 (1971), 50–59. D. Singerman. On the structure of non-Euclidean crystallographic groups. Proc. Camb. Phil. Soc. 76 (1974), 233–240. T. W. Tucker. Symmetric embeddings of Cayley graphs in non-orientable surfaces. In Graph theory, combinatorics and applications (ed. I. Alavy et al.), vol. 2, pp. 1105–1120 (Wiley Interscience, 1991). H. C. Wilkie. On non-Euclidean crystallographic groups. Math. Z. 91 (1966), 87–102.0308-210510.1017/S0308210507000169https://hdl.handle.net/20.500.14352/50033Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric cross-cap number and denoted by ˜σ(G). This number is related to other parameters defined on surfaces as the symmetric genus and the strong symmetric genus. The systematic study of the symmetric cross-cap number was begun by C. L. May, who also calculated it for certain finite groups. Here we obtain the symmetric cross-cap number for the groups Cm ×Dn. As an application of this result, we obtain arithmetic sequences of integers which are the symmetric cross-cap number of some group. Finally, we recall the several different genera of the groups Cm × Dn.engjournal articlehttp://journals.cambridge.org/abstract_S0308210507000169http://www.cambridge.org/restricted access512.54Automorphism groupsKlein surfacescross-cap numbersGrupos (Matemáticas)