Let $G \leq S_n$ be a permutation group of order $p^a q^b$, where $p$ and $q$ are primes. Here the group $G$ is may or may not be transitive.
The natural action of $G$ on $\{1,\ldots,n\}$ is defined as follows: $\chi:G \times \{1,\ldots,n\} \rightarrow \{1,\ldots,n\}$, $\chi(g,i)=i^g$.
The subgroup Point stabilizer of $i$ is defined as follows: $G_i=\{g \in G \mid i^g=i\} \leq G$.
Question: What are these point stabilizer subgroups $G_i$ of $G$ ?
(That is, are the subgroups $G_i$ well studied?)
Refer this links for the definition of stabilizer and transitive