circuit of $F_4^\dagger$ is the standard circuit for Fourier transforms [@Nielsen]. Note that the first permutation of the qubits can be removed when we change the order of the input.

Dihedral groups {#Section 2.2}
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The cyclic symmetry group of an $m$-sided regular polygon which we considered in the previous section is a subgroup of the dihedral group. The dihedral group consists of all rotations which map the $m$-sided regular polygon onto itself. In contrast to the cyclic group we allow the rotations to have different axes. For a fixed $m\geq 2$, the dihedral group is isomorphic to $D_m=\langle r,s\rangle$ with $r^m=1$, $s^2=1$, and $srs=r^{-1}$. In order to use the results for the cyclic groups, we consider the same orientation of the regular polygon as in the previous section, i.e., the face of the polygon is orthogonal to the $z$-axis. Furthermore, we assume that at least one vertex is an element of the $x$-axis. Due to this orientation, the element $r$ corresponds to the $2 \pi/m$ rotation about the $z$-axis and the element $s$ corresponds to the $\pi$ rotation about the $x$-axis. In the Hilbert space these rotations correspond to the matrices $$\left(\begin{array}{cc}1 &0 \\ 0& \omega \end{array} \right) \; 