Why is it that $C_s$ character tables always list a $\sigma_h$ operation despite the definition of $C_s$ not necessitating the reflection plane be horizontal?

In fact, how can the reflection plane even be characterised without a principal rotation axis to base it on?

Schoenflies Flowchart Character Table


There is always a principal rotation axis, even if it's only a $C_1$ axis. As there are an infinite number of these one can always be found perpendicular to the reflection plane, and as horizontal planes take precedence over veritcal ones, the operation is listed as $\sigma_h$.

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