Other Translations and Mirror Lines

Remember $\tau$ is viewed as translation right by one unit. Then translation by all whole numbers of units left or right are also in the symmetry group. In fact since $\tau$ was a translation by the smallest distance in our symmetry group, there can't be any other translations. For example, if translation 3.7 units right were in our group, then following this by translation 3 units left would give us a translation by .75 units - too small.

That said, we get a restriction on which vertical mirrors are possible. Suppose our symmetry group has a reflection r with mirror along the y axis, and another reflection with mirror x=c. The composition of these two mirrors is one of our translations. So what are the possible values of c?