# Character table and symmetry operations

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?

$$S_{\infty}$$ is equivalent to $$\sigma_\mathrm{h}$$. $$S_{\infty}$$ means rotate 0° then reflected by $$\sigma_\mathrm{h}$$, which is just same as taking $$\sigma_\mathrm{h}$$ itself.