I'm thinking about the fundamental group of a circle with some points identified. I mean let $r:\mathbb S^1\to \mathbb S^1$ be a quotient map mapping the point of the circle $(cos \theta, sin \theta )$ to $(cos(\theta+2\pi /n),sin (\theta+2\pi /n))$. Form a quotient space identifying $x$ to $r(x), r^2(x), \ldots,r^{n-1}(x)$.
I need help to find this fundamental group.
Thanks