Fourier Transform and Dihedral Group While preparing for a calculus exam we noticed that the Group generated by the Fourier transform and complex conjugation, as a subset of the operators on the Schwartz space $\mathcal{S}$ of rapidly decaying smooth functions $f:\mathbb R^n \to \mathbb C$, is isomorphic to the Dihedral group $D_4$ of order 8.
For $f \in \mathcal{S}$ define
$$ \hat f := Ff := (2\pi)^{-\frac{n}{2}} \int_{\mathbb{R}^n} \exp(-ixy) f(y)dy$$
where $xy$ is the standard scalar product of $x$ and $y$.
We already know that $F$ is a bijection on $\mathcal{S}$ and
$$ \check f := F^{-1}f = F^3 f = (2\pi)^{-\frac{n}{2}} \int_{\mathbb{R}^n} \exp(ixy) f(y)dy$$
If we now define $G = \langle F, \bar\cdot \rangle$, then $\mathrm{ord\:} F = 4$ and $\mathrm{ord\:}\bar\cdot = 2$. For $f \in \mathcal{S}$ and $x \in \mathbb{R}^n$ we also get the equation 
\begin{align}
F\bar f(x) &=  (2\pi)^{-\frac{n}{2}} \int_{\mathbb{R}^n} \exp(-ixy) \bar f(y)dy\\
&= (2\pi)^{-\frac{n}{2}} \int_{\mathbb{R}^n} \overline{\exp(ixy) f(y)}dy\\
&=\overline{F^{-1}f}(x)
\end{align}
So we also got the nessessary relation between the "flip" $\bar\cdot$ and "rotation" $F$. Thus $G$ is isomorphic to $D_4$. If we furthermore define 
$$ \tilde f(x) := \overline{f(-x)} = \overline{F^2f}(x)$$
we get some helpful equations (please excuse the abuse of notation)
$$\hat{\bar f} = \bar{\check f} = \tilde{\hat f} = \check{\tilde f}$$
$$\bar{\hat f} = \check{\bar f} = \hat{\tilde f} = \tilde{\check f}$$
From $G\cong D_4$ it follows that $\langle F^2 \rangle$ is the center of $G$ and the Hasse diagram of the lattice of subgroups of $G$ looks like this https://i.stack.imgur.com/zssSG.png. 
Now to my question:

Are there any other interesting statements we could obtain from $G$ being isomorphic to $D_4$? And do you know of similar links between abstract algebra and calculus (or more specifically the Fourier transform) ?

 A: There are many connections between "(abstract) algebra" and "calculus/analysis", often by groups of symmetries.
For example, if a function on $\mathbb R^n$ is rotationally invariant, then its Fourier transform is also rotationally invariant (you can prove this by change-of-variables in the definition of Fourier transform).
As another example, if a "function" (ok, tempered distribution) $u$ on $\mathbb R^n$ is (positive) homogeneous of degree $s$, meaning that $f(cx)=c^s\cdot f(x)$ for $c>0$, then its Fourier transform (as tempered distribution) is homogeneous of degree $-(s+n)$.
As a slightly different type of example, the possible symmetries of functions on spheres (keyword "spherical harmonics") are exactly understood in terms of the irreducible representations of the corresponding group of rotations (an orthogonal group).
There are yet-fancier examples, too: to make models for quantum mechanics, among other possibilities people have considered the differential operators on $\mathbb R$ with polynomial coefficients. Inside this collection of operators are operators which give a "representation" of the Lie algebra of the group $SL(2,\mathbb R)$ of two-by-two invertible real matrices.
On-and-on... :)
