In his paper from 2012, Nicolas Thiéry gives a formula for computing the Cartan matrix of a finite monoid (which can be seen as a measure of how "not semisimple" the algebra over the monoid is) in terms of number of fixed points under a conjugacy like action and the character table of the monoid. Motivated by the implementation of this formula, we will present in a first part an algorithm (and more importantly the results on which it rests) to efficiently compute this number of fixed points in finite monoids. Secondly, we will present a formula for the character table of the monoid that reuses these fixed points considerations to speed up the computation, culminating in an implementation of the computation of both the character table and the Cartan matrix. As our main objective is to contribute a computational tool to the monoid community, we will conclude by remarks on performance and practical use.