I'm studying for exams and came across this problem
My attempt
I'm going to use van Kampen. Label the bottom horizontal edge $c$ and the bottom right diagonal edge $d$. For our path connected open sets I'm going to choose $U$ to be an open disc inside $Y$, and $V$ to be $Y$ minus a point inside $U$. $U,V, U \cap V$ are all path connected with the latter being an annulus around the point.
Computing the Fundamental Groups:
$\pi_1(U)$ is trivial, as $U$ is an open disc so it's contractible.
$\pi_1(U\cap V) \cong \mathbb{Z} = \langle b| \emptyset \rangle$ as the annulus deformation retracts to $S^1$.
Question: How do I find the Fundamental Group of $V$? I know that $V$ deformation retracts to just the boundary of this hexagon, so only the vertices and edges, but I can't figure out what the Fundamental Group of this boundary is.
Once I have this the problem should be simple: Using van Kampen $$\pi_1(Y) \cong \langle \text{generators of $\pi_1(V)$}| \text{relations of $\pi_1(V)$}, i_1(b) = i_2(b) \rangle$$ Where $$i_1:\pi_1(U\cap V) \rightarrow \pi_1(U)$$ $$i_2:\pi_1(U\cap V) \rightarrow \pi_1(V)$$ $i_1$ is the trivial map, so the generator of $\pi_1(U\cap V)$ is sent to 1.
Going off of previous van Kampen problems I think $i_2: r\mapsto cdb^{-1}aab^{-1}$, where the generator of the $\pi_1(U\cap V)$ going once around the circle in the intersection, but goes around the boundary of the hexagon once. This can just be read off of the identification of the boundary. In which case I would get
$$\pi_1(Y) \cong \langle \text{generators of $\pi_1(V)$}| \text{relations of $\pi_1(V)$}, cdb^{-1}aab^{-1} =1 \rangle$$
All that is left is to find the generators and relations of $\pi_1(V)$, but I'm stuck.