TITLE: The largest group acting on a non-orientable surface of genus 5
QUESTION [6 upvotes]: Let $N_5$ denote the non-orientable surface of genus 5.
In Conder's database https://www.math.auckland.ac.nz/~conder/BigSurfaceActions-Genus2to101-ByGenus.txt we can see that the biggest finite group $F$ acting on $N_5$ has order 120. Moreover, the quotient has signature $(0; +; [-]; \{(2,4,5)\})$.
Is there a very concrete description of this group $F$?
To be even more concrete. I would like to the length $n$ of the largest chain of subgroups $1=F_0<F_1<\cdots< F_n=F$ of $F$. Since $120=(2^3)(3)(5)$ then $n\leq 5$. Is it $n=5$? Is $n$ strictly less than 5?
Thank you!

REPLY [9 votes]: The group $F$ is isomorphic to the symmetric group $S_5$.
In fact, since $N_5$ is non-orientable of genus $5$, both $F$ and the extended group $F^*$ (of order twice the order of $F$) act on its orientable double cover, that has genus $4$. In Conder's database, this is expressed by saying that the action of $F$ in genus $4$ is reflexible and that there is a non-orientable quotient.
Looking at the actions of groups of order $120$ on orientable surfaces of genus $4$ that have these properties, we can find the presentation of $F$; in fact, there is a unique such a group:

Now a simple computation with GAP4 does the job.