# 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?

• Have you tried rotating the crystal around to put the $a$ or $b$ axis perpendicular to the plane of observation? Maybe one of those directions has the $C_6$ in your structure. Dec 11 '19 at 7:52
• I didn't consider the axis along other direction because the lattice constant $c$ can be changed independently from $a$ and $b$. Thus if there exists a 6-fold rotation axis which is not along the $z$ direction, the rotation would mix the $z$-coordinate with $x$ and $y$. Such a rotation should not be preserved if we arbitrarily change the $c$ value, I think. Dec 11 '19 at 8:12
• your logic doesn't make sense to me, axes are arbitrary. I would try looking at those other axes instead of ignoring them. Dec 11 '19 at 8:21
• Let me put it this way: the point group is $C_{6h}$, which means there is a mirror plane perpendicular to the rotation axis. And in this system, the only presented mirror plane is the $xy$-plane, thus the corresponding 6-fold axis should be along the $z$ direction. Dec 11 '19 at 9:06
• Which material is this? Can you link to a cif or crystallography database entry? Dec 11 '19 at 9:18

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.
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)}.