Quasi-dihedral groups and generalized quaternion groups I am working on finding some properties of finite groups. My question is:                          are the elements / structure ( particularly the orders of elements ) of generalized quaternion groups, quasi-dihedral groups are well known as in the dihedral group ? And if so what are the references that I should look in ?
 A: The most basic properties of the groups dihedral, quaternion, and quasi-dihedral groups  are the following:
(1) the only non-cyclic $2$-groups with a cyclic subgroup of index $2$ are these only. 
(2) If $p=2$ and $G$ is a $p$-group with $[G:G']=p^2$ then $G$ is one among three above, and each of the above satisfies this property. (For $p>2$, the classification is very difficult and not known yet! But in case $p=2$, the classification shows that the class of such groups is very very small!)
(3) The only $2$-groups of maximal class are these three. 
The details can be found in Suzuki- Group Theory, or Huppert's  Finte groups (German) Vol.1 or Groups of prime power order Vol. 1 (Berkovich-Zanko).
However, for very specific information, such as you want, the only way could be do yourself the computation using above structural information or using presentation of them given in mentioned books. Since these groups are generated by just 2 elements, and large number of elements commute, the calculations you are expecting are easy to do.
