An important consideration of symmetry is that translational symmetry, where atoms are offset from their original positions by a fixed amount, will cause classes of reflections to have zero intensity, known as systematic extinctions or systematic absences. Only translational symmetry operations (cell centering, screw axes and glide planes) can cause systematic extinctions, while the presence of proper and improper rotation operations do not cause systematic extinctions except in combination with translational symmetry.
To understand how systematic extinctions arise, let us consider a unit cell with a C-center. To simplify we will assume that the unit cell has only two atoms, if the first is at x,y,z then the second will be at x+1/2, y+1/2, z. If take the structure factor equation,
\[F_{hkl}=\sum _j^N f_j \exp [-2\pi i(hx_j + ky_j + lz_j)]\]
and expand this over our two atoms, we get
\[F_{hkl} = f_j \exp [-2\pi i(hx_j + ky_j + lz_j)] f_j + \exp [-2\pi i(hx_j + ky_j + lz_j) + \frac {h+k}{2}] \]
which can be rewritten as
\[ F_{hkl} = f_j \exp [-2\pi i(hx_j + ky_j + lz_j)] f_j + \exp [-\pi i(h + k)] \exp [-2\pi i(hx_j + ky_j + lz_j) ] \]
Note that when h + k is even, then \(\exp [-\pi i(h + k)] = 1\) but when h + k is odd then \(\exp [-\pi i(h + k)] = -1\) so the sum of the two exponentials is zero. Thus, for C-centering with any number of atoms in the unit cell, \(F_{hkl}=2\sum _j^{N/2} f_j \exp [-2\pi i(hx_j + ky_j + lz_j)]\) for h+k even, and \(F_{hkl}=0\) for h + k odd.
We can also rationalize this by thinking that if viewed along a reciprocal space direction, symmetry causes the unit cell repeat distance to decrease by a half, then half of the reflections in that projection must disappear. For a 21 screw axis along a, the viewed along the a*-axis, the cell will appear to be half the original size. Thus, h00 reflections will be absent when h is odd. This only occurs in projection along a*. From any other direction this superposition is broken. Likewise, for a c-glide plane perpendicular to the b-axis, this appears to halve the c-axis length except when not perpendicular to b*, so h0l reflections must have zero intensity when l is odd.