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!