How many times is each alternating group a hurwitz quotient? In other words, how many non-isomorphic ways can an alternating group be generated by an element of order 2 and an element of order 3 whose product has order 7? Specifically, how many times do $A_{15}$, $A_{21}$, $A_{22}$, $A_{28}$, $A_{29}$, and $A_{30}$ appear as hurwitz quotients?
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$\begingroup$ In case you are not aware of it let me mention the paper of Liebeck and Shalev, "Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks" which studies a generalisation of your question. They give formulas for the number of permutation representations, and though they are mostly interested in asymptotic aspects it might help also for explicit counting problems (I'm too lazy to do it myself so this is very far from an answer). $\endgroup$– Jean RaimbaultCommented Feb 14, 2018 at 8:12
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