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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.

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  • $\begingroup$ Hmm, so this group is an extension of the dihedral group $\langle a^2, b\rangle$ of order $2^m$ by $\mathbb Z/2\mathbb Z = \langle a\rangle/\langle a^2\rangle$. I'm not sure if there's some Mackey machinery that could do the job …. $\endgroup$
    – LSpice
    Commented Aug 24, 2018 at 15:02
  • $\begingroup$ I do not know this machinery, but I'll give it a look. I study group actions on a smooth curve and consequently on their Jacobian. As I study, I would need these representations. Thank you @LSpice $\endgroup$
    – Kelyane Abreu
    Commented Aug 24, 2018 at 15:08
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    $\begingroup$ These are generally called semidihedral groups. The irreducible representations all have degree $1$ or $2$, four of degree $1$, and $2^{m-1}-1$ of degree $2$, where the latter are induced from the irreducibles of the subgroup $\langle a \rangle$. $\endgroup$
    – Derek Holt
    Commented Aug 24, 2018 at 15:13
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    $\begingroup$ This is also a semidirect product $(Z/16Z)\rtimes C_2$ where the action of the generator of $C_2$ is by multiplication by 7 [$(Z/2^mZ)\rtimes C_2$ with multiplication by $2^{m-1}-1$ in the general case]. From this it's quite clear how to describe irreducibles by hand, by first decomposing over $Z/16Z$, etc, and get Derek's description; you should do it as an exercise. $\endgroup$
    – YCor
    Commented Aug 24, 2018 at 15:59
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    $\begingroup$ The $2$-dimensional irreducible representations are $a \mapsto \left(\begin{array}{cc}\omega&0\\0&\omega^7\end{array}\right)$, $b\mapsto \left(\begin{array}{cc}0&1\\1&0\end{array}\right)$, where $w$ is a $16$th root of $1$, $w \ne \pm 1$. $\endgroup$
    – Derek Holt
    Commented Aug 24, 2018 at 16:21

2 Answers 2

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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.

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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 $-1$.

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