Quick dihedral question

Hi for a dihedral group where r represents a rotation and s a reflection what does rs mean?

Is it a rotation followed by a reflection?

Or a reflection followed by a rotation?

Thanks!

also is the rotation r clockwise or anticlockwise?

We consider them to be like functions acting on the polygon. Hence $rs$ means "perform s, then r" - that is, it's the reflection and then the rotation. Much like $fg(x)$ is f applied to $g(x)$.



It doesn't really matter - just pick a direction and stick to it. I'm not even aware of a convention about this, because it turns out to be quite rare in practice that the orientation of the rotation matters. After all, if you define it one way, I can always look from the "other side of the table" (picture: you've drawn the shape on a glass table with rotation defined clockwise; I crawl under the table and look up, and see the same shape with the same transformations but with rotation defined anticlockwise).

Also, if you were asked to find the conjugacy classes for a group, say S3, is there a quick way to do this? Do you find the class for each elements and perform all the cycle multiplications (because this takes me forever)?

There is a theorem about this. Do you know the theorem that "$K$ is a conjugacy class in $S_n$ iff all elements of $K$ share the same cycle type and no other elements from $S_n - K$ have that cycle type"?