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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?

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$S_{\infty}$ 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.

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