  We make a systematic study of symmetry-protected topological gapped phases of
quantum spin chains in the presence of the frieze space groups in one dimension
using matrix product states. Here, the spatial symmetries of the
one-dimensional lattice are considered together with an additional 'vertical
reflection', which we take to be an on-site $\mathbb{Z}_2$ symmetry. We
identify seventeen distinct non-trivial phases, define canonical forms, and
compare the topological indices obtained from the MPS analysis with the group
cohomological predictions. We furthermore construct explicit renormalization
group fixed-point wave functions for symmetry-protected topological phases with
global on-site symmetries, possibly combined with time-reversal and parity
symmetry. En route, we demonstrate how group cohomology can be computed using
the Smith normal form.
