Geodesic
The symmetric genus spectrum of finite groups
Ars Mathematica Contemporanea, Tome 4 (2011) no. 2, pp. 271-289.

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The symmetric genus of the finite group G, denoted by σ(G), is the smallest non-negative integer g such that the group G acts faithfully on a closed orientable surface of genus g (not necessarily preserving orientation). This paper investigates the question of whether for every non-negative integer g, there exists some G with symmetric genus g. It is shown that that the spectrum (range of values) of σ includes every non-negative integer g =!= 8 or 14 mod 18, and moreover, if a gap occurs at some g == 8 or 14 modulo 18, then the prime-power factorization of g − 1 includes some factor pe == 5 mod 6. In fact, evidence suggests that this spectrum has no gaps at all. 
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Marston D. E. Conder; Thomas W. Tucker. The symmetric genus spectrum of finite groups. Ars Mathematica Contemporanea, Tome 4 (2011) no. 2, pp. 271-289. doi : 10.26493/1855-3974.127.eb9. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.127.eb9/

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