In group theory, the quaternion group $Q_8$ (sometimes just denoted by $Q$) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication.
Question : How can I understand this group? Please explain simply. I have read its definition on Wikipedia but I did not get more than that it is non-abelian group of order eight. I am looking for simplest representation of it. What is the underlying operation? How many elements of order two, four, eight are there?