Let $G \cong \Bbb{Z}_q \rtimes Q_8$. Then $G$ has a presentation as follows
$$\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$$
I dont understand why $x^z=x^{-1}$?
(The action is conjugation)
2026-03-26 20:16:42.1774556202
Showing $\Bbb{Z}_q \rtimes Q_8$ has the presentation $\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$
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