Quandles of cyclic type with several fixed points

A quandle of cyclic type of order $n$ with $f\geq 2$ fixed points is such that each of its permutations splits into $f$ cycles of length $1$ and one cycle of length $n-f$. In this article we prove that there is only one such connected quandle, up to isomorphism. This is a quandle of order $6$ and $2$ fixed points, known in the literature as octahedron quandle. We prove also that, for each $f\geq 2$, the non-connected versions of these quandles only occur for orders $n$ in the range $f+2 \leq n \leq 2f$ and that, for each $f>1$, there is only one such quandle of order $2f$ with $f$ fixed points, up to isomorphism. Still in the range $f+2 \leq n \leq 2f$, we present sufficient conditions for the existence of such quandles, writing down their permutations; we also show how to obtain new quandles form old ones, leaning on the notion of common fixed point.