# The covering space of a bouquet of 2 circles corresponding to a normal subgroup

Consider $S_3$ with this presentation: $S_3=\left\langle\sigma,\tau:\sigma^2=1, \sigma\tau=\tau^{-1}\sigma\right\rangle$. Let F be the free group with two generators $s,t$ and $R$ the minimal normal subgroup of $F$ containing $s^2$ and $sts^{-1}t$. What is the covering space of the bouquet of 2 circles corresponding to $R$?

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Since you've recognized $F/R$ as isomorphic to $S_3$, you already know how large that covering space is, meaning what is the order of the covering (what is it?), and you also know what permutations define the paths that lie over each of the circles in the bouquet (they correspond to $\sigma$ and $\tau$). Can you wrap this up from here?