I am going back through Dummit/Foote studying for a prelim and came across the 'quasidihedral' or 'semi-dihedral', group of order $2^n$, with presentation $\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}-1}\rangle$. I was wondering if there is a representation of these groups, maybe even just for small order, that is feasible to visualize. I tried to find such online without any luck.
There is the very similar $M_n(2) = \langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}+1}\rangle$ which I am also curious about. In particular I would be curious to know if there is a way to derive a vague understanding of the geometry of this group from the similarly-presented quasidihedral groups above.
Thanks!
The geometry of these groups is best seen through their respective actions on the Cayley graph. The following animation shows the left action of $r$ on the Cayley graph of $M_5(2)$:
Black edges correspond to right multiplication by $r$, and should be oriented counterclockwise on both circles. Red and blue edges correspond to right multiplication by $s$. The left action of $s$ is a reflection across a horizontal plane that flips both circles.
The Cayley graph of the quasidihedral group of order 32 is actually the same graph, with the same left action of $r$. The only difference is that the black edges on one of the two circles should be oriented clockwise, and left-multiplication by $s$ acts as a $180^\circ$ rotation around a horizontal axis instead of a reflection.
This "two circles rotating at different rates" geometry is reflected in the linear representations. The quasidihedral group of order $2^n$ has a faithful two-dimensional complex representation: $$ r \mapsto \begin{bmatrix}\omega & 0 \\ 0 & -\overline{\omega}\end{bmatrix},\qquad s\mapsto \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, $$ where $\omega$ is a primitive $2^{n-1}$ root of unity. Note that $r$ acts as a rotation of each component of a vector, while $s$ switches the two components of a vector. Ignoring the complex structure leads to a faithful four-dimensional real representation.
The group $M_n(2)$ has a similar faithful two-dimensional complex representation: $$ r \mapsto \begin{bmatrix}\omega & 0 \\ 0 & -\omega\end{bmatrix},\qquad s\mapsto \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, $$ Again, this leads to a four-dimensional real representation.