I learned that a subgroup of $D_n = \langle r,s \mid r^n=s^2=(rs)^2=1 \rangle$, the dihedral group of order $2n$, is either cyclic or dihedral itself, and that a subgroup of the latter kind is of the form $\langle r^d, sr^i\rangle =: G(d, i)$, where $d\mid n$ and $0\le i < d$.
I would like to interpret this fact geometrically. I feel that $G(d,i)$ is the set of congruent transformation that preserves the regular $n/d$-gon that one gets by picking every $d$-th vertex of the original $n$-gon that the elements of $D_n$ preserves. However, in this picture I do not know how to interpret the generator $sr^i$ of order two of $G(d,i)$; since it is of order 2, I think it is a rotation about some axis, but I am not sure whether it preserves the $n/d$-gon in question.
I would be grateful if you could give some geometric intuition about this matter.