So, we know that in a Dihedral Group $D_n$, if $r$ represents counterclockwise rotation by $2\pi/n$ radians and $s$ is any axis of reflection, then the elements of reflection stand as follows:
{$s, rs,r^2s, ...... , r^{n-1}s$} .
In the image attached, there is an octagon . Suppose i consider the reflection axis $'s'$ as $ab$. Now, since $r$ stands for rotation by a degree of $360/8 = 45^o$, $rs$ must rotate $s$ by $45^o$ to obtain the $rs$ as $cd$. But, $rs$ is shown to be the axis passing through vertices $3$ and $7$ . Why is this paradox happening?
Instead of us calculating where vertices 3 and 4 go individually considering $r$ and $s$ , lets consider the cumulative effect of $rs$. Since, we have fixed the axis s, now, rs means, rotate this $s$ axis by $r$ radians in counter clockwise direction and then, since $rs$ is a reflection, reflect the original figure about the new axis described by $rs$.
Thank you