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Consider the following paragraph from a book of Magnus on Combinatorial Group Theory.

... the simple group $G_{168}$ of order $168$ acts on a genus $3$ surface, is important for the theory of equations of degree $7$ with Galois group $G_{168}$. It is impossible to develop the theory of these equations in the same manner as the theory of general equations of fifth degree with Galois group $S_5$, since the automorphic functions belonging to a Riemann surface of positive genus are not simply rational functions of a single automorphic function

I didn't understand the italic statement above, especially how the impossibility is convinced using automorphic functions? I didn't understand this jumping from groups to automorphic functions. Can one explain a-little-bit this para?

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