The following is a past qual problem: let $X = S^1 \times [0,1]$ be the cylinder, and define an equivalence relation on $X$ by $(z,1) \sim (iz,1)$ and $(w,0) \sim (e^{i\pi /7} w, 0)$. Compute $\pi_1(X/{\sim})$.
I've been trying to realize a CW complex structure for $X/{\sim}$, by having a 2-cell with edges $a^4$ at one end and a 2-cell with edges $b^7$ at the other and then attaching them somehow (of course, here we have one vertex and loops labelled by $a$ and $b$).
Moreover, can we realize $X/{\sim}$ as the pushout of a diagram $Y_1 \stackrel{f}{\longleftarrow} S^1 \stackrel{g}{\longrightarrow} Y_2$, where $f,g$ would be quotient maps by different rotations?
Hint: Do both problems separately, where you do only one of the two identifications (for example only $(z,1) \sim (iz,1)$, or only $(w,0) \sim (e^{i\pi /7} w, 0)$). Then combine the two results using the Seifert-van-Kampen theorem.