Symmetry structures of tilings on a Klein bottle

This study presents a comprehensive characterization of the symmetry structures of tilings on a Klein bottle, arising from tilings of the Euclidean plane with crystallographic symmetry groups {\cal G} containing a subgroup {\cal L} of type pg. The investigation focuses on determining the isometries in the normalizer group {N_{\cal G}}\left({\cal L} \right) through diagrams of isometries, utilizing subgroup relationships among plane groups to streamline computations. A key result reveals that the quotient group {N_{\cal G}}\left({\cal L} \right)/{\cal L} can be decomposed as a product of cyclic and dihedral groups, with its order depending solely on the power of the generating translation of {\cal L} in the direction of the glide reflection axes.