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A method to do this for the group $\textrm{PSL}_2 (\mathbb{F}_{p^n})$ can be found in the papers by Glover and Sjerve:

Representing $PSl_2(p)$ on a Riemann surface of least genus, L'Enseignement Mathématique 31 (1985)

The genus of $PSl_2(q)$, Journal für die reine und angewandte Mathematik 380 (1987).

A method to do this for the group $\textrm{PSL}_2 (\mathbb{F}_{p^n})$ can be found in the papers by Glover and Sjerve:

Representing $PSl_2(p)$ on a Riemann surface of least genus, L'Enseignement Mathématique 31 (1985)

The genus of $PSl_2(q)$, Journal für die reine und angewandte Mathematik 380 (1987).

A method to do this can be found in the papers by Glover and Sjerve:

Representing $PSl_2(p)$ on a Riemann surface of least genus, L'Enseignement Mathématique 31 (1985)

The genus of $PSl_2(q)$, Journal für die reine und angewandte Mathematik 380 (1987).

A method to do this for the group $\textrm{PSL}_2 (\mathbb{F}_{p^n})$ can be found in the papers by Glover and Sjerve:

Representing $PSl_2(p)$ on a Riemann surface of least genus, L'Enseignement Mathématique 31 (1985)

The genus of $PSl_2(q)$, Journal für die reine und angewandte Mathematik 380 (1987).

A method to do this can be found in the papers by Glover and Sjerve:

Representing $PSl_2(p)$ on a Riemann surface of least genus, L'Enseignement Mathématique 31

The genus of $PSl(2,q)$, Journal für die reine und angewandte Mathematik 380.

A method to do this can be found in the papers by Glover and Sjerve:

Representing $PSl_2(p)$ on a Riemann surface of least genus, L'Enseignement Mathématique 31 (1985)

The genus of $PSl_2(q)$, Journal für die reine und angewandte Mathematik 380 (1987).