TITLE: Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$
QUESTION [20 upvotes]: The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)?

The best I could get by trial and error is an embedding without crossings on a sphere with 10 crosscaps.
Running the following in Sage works in theory, but takes too long:

A5 = groups.permutation.Alternating(5)
S = [(1,2,3,4,5),(1,4,3,2,5)]
d = A5.cayley_graph(generators=S)
bt = d.to_undirected()
bt.genus()

In Sage we can also use JSMol to get a not-so-helpful view of the graph.

REPLY [2 votes]: According to Sagemath documentation, the time complexity of their algorithm is
$$
\mathcal{O}\left(|V| \prod_{v \in V} (d(v) - 1)!\right).
$$
(Note that in this instance this evaluates to $6^{60}$.)
However, quick google search reveals that approach via integer linear programming or SAT solvers might be viable for this particular case. I suggest you contact the authors of the following articles:
Stronger ILPs for the Graph Genus Problem, 27th Annual European Symposium on Algorithms (ESA 2019)
A Practical Method for the Minimum Genus of a Graph: Models and Experiments,  International Symposium on Experimental Algorithms, SEA 2016: Experimental Algorithms, pp 75-88