Given the generalized quaternion group $Q_{2^t} = \langle a,b : a^{2^{t-1} } = e, b^2 = a^{2^{t-2}}, b^{-1}ab = a^{-1}\rangle$, I have been asked to find it's (linear complex) representations.
My first idea was to find the conjugacy classes of $Q_{2^t}$. I did this successfully, getting the following classes : $[e], [a^{2^{t-2}}], [a^{\pm k}], [b],[ab]$, where $ 1 \leq k \leq 2^{t-2} - 1$. This gives $2^{t-2} + 3$ conjugacy classes in total.
From here, a bunch of facts allowed me to conclude this: There are exactly $4$ one-dimensional representations, these are given by $a,b \to \pm 1$. (Each combination is possible). All the others are exactly two dimensional.
Now, the question is, how do I describe these?(as in, write these as matrices or find their characters). Then I had this question.
QUESTION : Note that $Q_{2^t} = \frac{\frac{\mathbb Z}{2^{n-1}} \rtimes \frac{\mathbb Z}{4}}{\langle 2,2^{n-2}\rangle}$, where the semi direct product is how it is defined in the representation of $Q_{2^t}$.
I can find the representations of the semi-direct product : it comes under Wigner's little groups method. Once I do this, can I obtain all representations of $Q_{2^t}$ by restriction, or taking a quotient of some sort? Or must I proceed by intuition i.e. guessing from the presentation of $Q_{2^t}$ what might be the matrices corresponding to those $2$ dimensional representations?