# 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 $$D_{\infty h}$$ point group table and it lists $$C_\infty$$, $$\sigma_v$$, $$S_\infty$$, $$i$$, and $$C_2$$. But doesn't it also have a $$\sigma_h$$? Why is this not included?

Am I missing something here?

In the $$D_{\infty \mathrm{h}}$$ point group, the $$\sigma_\mathrm{h}$$ reflection is equivalent to the $$2S_{\infty}(\phi)$$ improper rotation with the rotation angle $$\phi$$ equal to zero (recall that the definition of a $$C_n$$ symmetry element is a rotation through $$(360^\circ/n)$$ degrees).
• Ah I didn't see that. Perhaps I'm being lazy and should be a different question, but why the "2" on "$2S_\infty$? – ralk912 Sep 4 at 2:35
• I’m not sure whether it’s to do with the fact that you can rotate in either direction. Maybe ask a new question ;) I’m starting to doubt my answer, in fact, because the $C_n$ operation with $\phi = 0$ is also equivalent to $E$, and $E$ is one of the operations in the character table, so why not $\sigma_\mathrm h$. These infinite point groups are rather confusing... – orthocresol Sep 4 at 10:29
• I assume you are right about you explanation and that the reason they separate out $E$, despite it technically being redundant, is that it makes explicit that $D_{\infty h}$ is a group, with the required identity operator. Why that is more important than making explicit that the group actually has a $\sigma_h$ operation is up to the judgement of whoever printed these tables. – Tyberius Sep 4 at 15:56
• But also $i$, isn't $S_2$ equivalent to $i$? Or is $i$ particularly special too? – ralk912 Sep 5 at 18:01