It's been a while since I've dealt with these, but do character tables include all possible symmetry operations?
I am looking at the $D_\mathrm{\infty h}$ point group table and it lists $C_\infty$, $\sigma_\mathrm{v}$, $S_\infty$, $i$, and $C_2$. But doesn't it also have a $\sigma_\mathrm{h}$? Why is this not included?
Am I missing something here?