The background to this is that I've recently given a tutorial wherein we had to go through the determination of point groups for atoms in various lattices (BCC, FCC/CCP and HCP). BCC and HCP, I have no problem with. However, I've now acquired a copy of the lecturer's "official" solutions and there's a bit of discrepancy with the FCC, that I can't fathom: I get an $O_\mathrm{h}$ point group, while he gives a $D_\mathrm{3d}$ point group.
Here goes:
The coordination geometry of an atom in the FCC structure (assuming one atom/lattice point and only one species of atom, as in an FCC metal) is a cuboctahedron:
,
By my reckoning we have $3$ $C_4$ axes, one through each pair of square faces, and a $C_3$ through each pair of triangular faces.
Following the classic flowchart:
The number of high-symmetry rotational axes is surely enough to exclude $D_\mathrm{3d}$ already?
The only thing that makes me think that the lecturer hasn't just made an error is this line from the Wikipedia page for the cuboctahedron:
With $O\mathrm{h}$ symmetry, order $48$, it is a rectified cube or rectified octahedron (Norman Johnson).
With $T_\mathrm{d}$ symmetry, order $24$, it is a cantellated tetrahedron or rhombitetratetrahedron.
With $D_\mathrm{3d}$ symmetry, order $12$, it is a triangular gyrobicupola.
This is no doubt due to deficiencies in my understanding of group theory, but I don't understand how the same shape can simultaneously have multiple point groups, nor how you would select which one is relevant to a particular analysis.