Does the space group P63/m (No. 176) have C6 rotation symmetry?

The answer is that space group P6$_{3}$/m does have a corresponding point group of $C_{6h}$, and it does not have the symmetry of $C_6$.

This is not contradictory because P6$_{3}$/m is a non-symmorphic space group. A non-symmorphic group has at least one element that contain non-interger translation. If we put all pure translations into a group T, we can see that T is a subgroup of the space group G. And we can divide G by T to get a factor group F, which is a subgroup of G.

For a symmorphic group, the factor group F is nothing but its corresponding point group P, while for a non-symmorphic group, its point group is obtained by omitting the translation part of the factor group. In this case the point group is isomorphic to F, but it is not a subgroup of G.

Therefore, despite that $C_6$ is an element of point group $C_{6h}$, since $C_{6h}$ is not a subgroup of P6$_3$/m, $C_6$ is not an element of space group P6$_3$/m. The corresponding element in the factor group is the screw operation {$C_6$|(0,0,1/2)}.