Q-group

Definition with symbols
A group $$G$$ is termed a Q-group if it has a subgroup $$A$$ satisfying the following:


 * $$A$$ is Abelian but not elementary Abelian
 * $$A$$ is a subgroup of index two in $$Q$$, hence normal
 * There is an element $$x \in G \setminus A$$ such that $$x^2$$ is an element in $$A$$ of order two