Fundamental group of quotient of $S^1 \times [0,1]$

Question

I have a past qual question here: Let $X = S^1 \times [0,1] /{\sim}$, where $(z,0) \sim (z^4,1)$ for $z \in S^1 = \{ z \in \mathbb{C} \colon \| z \| = 1 \}$. Compute $\pi_1(X)$.

I've been trying to visualize $X$ as a cylinder of height 1 with the two ends identified `with a twist', but this has not seemed to help. Any help or hints would be much appreciated!

Answer 1

You can realize this space as a CW complex by attaching a $2$-cell to the wedge sum of two circles $\{a, b\}$ along the path $aba^{-4}b^{-1}$.