Hyperelliptic compact non-orientable Klein surfaces without boundary

Let X be a nonorientable Klein surface (KS in short), that is a compact nonorientable surface with a dianalytic structure defined on it. A Klein surface X is said to be q-hyperelliptic if and only if there exists an involution Φ on X (a dianalytic homeomorphism of order two) such that the quotient X/� Φ� has algebraic genus q. q-hyperelliptic nonorientable KSs without boundary (nonorientable Riemann surfaces) were characterized by means of non-Euclidean crystallographic groups. In this paper, using that characterization, we determine bounds for the order of the automorphism group of a nonorientable q-hyperelliptic Klein surface X such that X/� Φ� has no boundary and prove that the bounds are attained. Besides, we obtain the dimension of the Teichmuller space associated to this type of surfaces.