Does the space group P63/m (No. 176) have C6 rotation symmetry? Recently I'm working on a compound with space group P63/m. The top view of its structure is shown below (where only atoms of z=1/4 are shown).

From the list of space groups (Wiki: List of Space Groups), I found that the space group P63/m has a subgroup of $C_{6h}$, which means that the system should have $C_{6}$ rotation symmetry. However, by checking the previously mentioned structure, I cannot find such a $C_{6}$ rotation axis. 
Is there anyone who can help me to find out the "missing" $C_{6}$ rotation axis? Or did I misunderstand the symmetry of this system?
 A: 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)}.
A: You don't have simple (proper) 6-fold rotation axes, but (improper) 6-fold screw axes, that is a rotation followed by a translattion along the c-axis. In your picture only atoms at z=1/4 are shown; so you're missing the translation and you cannot see the screw axis. enlarge the z- range and you will see the 6-fold symmetry; look at the TlMo3Se3 below, the 6-fold axis is evident.

