A class of finite groups with Abelian centralizer of an element of order $3$ of type $(3,2,2)$

In this work, we study the structure of finite groups in which the centralizer of an element of order $3$ is isomorphic to $\mathbb Z_3\times\mathbb Z_2\times\mathbb Z_2$. The analysis is restricted to the class of groups whose order is not divisible by the prime number $5$. It is shown that among finite simple groups such groups do not exist, and a detailed possible internal general structure of such groups is investigated. We use only those results that have been published before 1980.