I'm right now learning about Monodromy from self-studying Rick Miranda's fantastic book "Algebraic Curves and Riemann surfaces". Today, I read about monodromy, and the monodromy representation of a holomorphic map between compact Riemann surfaces. I understand that we start by having a holomorphic map $F:X \rightarrow Y$, of degree d, where X and Y are Riemann surfaces, and then we remove the branch points from Y, and all the corresponding points in X mapping to them. Let $B=\{b_1,..,b_n\}$ be the branch points, $A=\{a_1,...,a_m\}$ the ramification points. So, fix a point $q \in V = Y-B$. We have that there are d preimages of q in $U=X-A$.
So for a specific branch point b, we choose some small open neighbourhood W of b so that $F^{-1}(W)$ gives a disjoint union $W_i$ of open neighbourhoods of the points mapping to b. Take some path from our basepoint $q$ to $q_0 \in W$, call this path $\alpha$. Choosing some small loop $\beta$ with basepoint q, with winding number 1 around b, and then considering $\alpha^{-1}\circ \beta \alpha$, gives a loop on V, based at q around b. We can now see that this loop only depends on $\beta$, in some sense.
Say that the points that maps to b has multiplicity $n_i,...,n_j$. Then we have that, according to local normal form, there are local coordinates $z_j$ on the open neighbourhoods from above, so that the map takes the form $z=z_j^{n_j}$. Now, we have that the loop around b, when we lift it up here, will simply yield a cyclic permutation of the preimages in the neighbourhood.
Now, my question is mostly: How do I apply this concretely? Let us take an example (from Miranda's book) : "Let $f(z) = 4z^2(z-1)^2/(2z-1)^2$ define a holomorphic map of degree 4 from $P^1$ to itself. Show that there are three branch points, and that the three permutations in $S_4$ are $\rho_1=(12)(34)$, $\rho_2(13)(24)$ and $\rho_3=(14)(23)$ up to conjugacy." I can find the branch points, and I see that the multiplicity of the two points mapping to it has multiplicity 2, but I don't get how to rigorously show that the above are the associated permutations.
Hope I was clear, and sorry if I wasn't.
UPDATE Now, rereading the question properly, maybe he doesn't want me to find the specific permutations, but just simply showing that they have that conjugacy class. I think that is the case. But I would still be curious of how to find the specific permutation that the monodromy induces.