In the paper by Ben Elias et al. (2010), Page 2, the minimal faithful permutation representation is defined.
Later it is mentioned that, choosing a point stabilizer subgroup in each orbit, it is clear that minimal faithful permutation representations corresponds to collections $\mathcal{H}$ of subgroup of $G$, which gives $d(G)$. Here, $d(G)$ is the least integer such that $G$ can be embedded in $S_d$.
Question: I did not get, how does some collection of point stabilizers gives a minimal faithful permutation representations of $G$.
E.g., Let $G =\langle (1,4)(2,5)(3,6),(1,3,2)(4,5,6) \rangle \leq S_6$. Then $G\cong S_3$ and it is transitive subgroup in $S_6$. Thus the orbit is $\Omega=\{1,2,3,4,5,6\}$. The action $\chi: G \times \{\Omega\} \to \{\Omega\}$ and $G_{\Omega}=\{e\}$. But the collection $\{G_{\Omega}\}$ does not give $d(G)$ (here $d(G)$ is 3).
I think I am not interpreting the definition correctly.
Can someone clear my doubt? Any help will be appreciated. Thanks.