Irreducible representations of $G_4 = \langle a,b \mid a^{16}, b^{2}, baba^{-7}\rangle$ and other Semidihedral groups I would like to know the irreducible representations of the group $G_4 = \langle a,b \mid a^{16}, b^2, baba^{-7}\rangle$   and its character table.
More than that, I would like to know the irreducible representations of the general group $G_m = \langle a,b \mid a^{2^{m}}, b^2, baba^{-d}\rangle$ where $d=2^{m-1}-1$.
I would be grateful if you could help me and indicate some references.
 A: For $m=4$ this may be useful:
https://people.maths.bris.ac.uk/~matyd/GroupNames/1/SD32.html
A few of the general cases are also here:
https://people.maths.bris.ac.uk/~matyd/GroupNames/semidihedral.html
The standard reference for ordinary finite group representations would be James and Liebeck's ``Representation and characters of groups".
In general the $2$-dimensional representations are:
$a\mapsto\begin{bmatrix}\omega&0\\0&\omega^{2^{m-1}-1}\end{bmatrix}$ and $b\mapsto\begin{bmatrix}0&1\\1&0\end{bmatrix}$
where $\omega$ is a $2^m$th root of unity that does not equal $1$ or $-1$.
A: There are good answers to this question both in formal answers and in the comments, but I'll make a couple of general remarks. If a finite group $G$ has a (necessarily normal)  Abelian subgroup $A$ of index $2,$ then any complex irreducible character $\chi$ of $G$ has degree at most $2,$ for by Clifford's Theorem, ${\rm Res}^{G}_{A}(\chi)$ is a sum of (necessarily degree $1$) irreducible characters of $A$. Note that $\langle {\rm Res}^{G}_{A}(\chi),{\rm Res}^{G}_{A}(\chi) \rangle_{A}\leq 2$ since $\langle \chi, \chi \rangle_{G} = 1.$ 
   Hence ${\rm Res}^{G}_{A}(\chi)$ is either irreducible, or is a sum of two distinct irreducible characters of $A$ (in the same $G$-orbit).
 Hence determining the irreducible characters of such  group boils down down to determining which irreducible characters of $A$ extend to $G$ ( ie are $G$-stable). The $G$-stable irreducible characters of $A$ each extend in two ways to an irreducible character of $G$ ( given any one extension, it does not vanish identically outside $A$, and if we multiply it by the linear character of $G$ with kernel $A$, we get a different extension). 
   Any irreducible character of $A$ which is not $G$-stable induces irreducibly to $G$, in the same way as the other irreducible character in its $G$-orbit.
 In your group, it is easy to determine which irreducible characters of $\langle a \rangle$ are $b$-stable (hence $G$-stable), so it is routine to determine the character table.
