  In this paper we present an algorithm for efficiently counting fixed points
in a finite monoid $M$ under a conjugacy-like action. We then prove a formula
for the character table of $M$ in terms of fixed points and radical, which
allows for the effective computation of the character table of $M$ over a field
of null characteristic, as well as its Cartan matrix, using a formula from
[Thi\'ery '12], again in terms of fixed points. We discuss the implementation
details of the resulting algorithms and provide benchmarks of their
performances.
