  A topological phase is a phase of matter which cannot be characterized by a
local order parameter. It has been shown that gapped phases in 1D systems can
be completely characterized using tools related to projective representations
of the symmetry groups. We show how to determine the matrices of these
representations in a simple way in order to distinguish between different
phases directly. From these matrices we also point out how to derive several
different types of non-local order parameters for time reversal, inversion
symmetry and $Z_2 \times Z_2$ symmetry, as well as some more general cases
(some of which have been obtained before by other methods). Using these
concepts, the ordinary string order for the Haldane phase can be related to a
selection rule that changes at the critical point. We furthermore point out an
example of a more complicated internal symmetry for which the ordinary string
order cannot be applied.
