The generalized quaternion group $\Bbb H_n$

Solution 1:

Yes. You got it right. All the elements of this group are of the form $$ \left(\begin{array}{cc}\omega^j&0\\0&\omega^{-j}\end{array}\right)\qquad\text{or}\qquad \left(\begin{array}{cc}0&-\omega^j\\\omega^{-j}&0\end{array}\right), $$ with $0\le j<2^n$.

A curious property of these groups is that they are the only non-cyclic $p$-groups with a fixed-point-free representation (i.e. a group of matrices such that the neutral element is the only one that has one as an eigenvalue). Consequently they play a role in Zassenhaus' classification of fixed-point-free groups and finite near-fields.

Solution 2:

This group can be defined "purely in group-theoretic terms":

$$ R^{2^n}=1, \ R^{2^{n-1}} = S^2, \ S^{-1}RS=R^{-1} $$ ($S^{-1}R=R^{-1}S^{-1}$ is unnecessary). This group is described in W.Burnside, Theory of groups of finite order, sect.105, Theorem VI.