Is there a non abelian group that characterize a one dimensional lattice structure?

Question

Of all groups that characterize a one dimensional lattice structure (symmetry operations including translation, $C_2$, mirror plane, inversion point), is there a non abelian one? Moreover, Can it have two dimensional irreducible representation?

Answer 1

The isometries of the real line are of the forms $t_a(x)=a+x$ and $r_a(x)=a-x$.

We have $$t_a\circ t_b = t_{a+b}\\ t_a\circ r_b = r_{b+a}\\ r_a\circ t_b = r_{a-b}\\ r_a\circ r_b = t_{a-b}$$

Also, $t_0$ is the identity.

They are not commutative: $$r_0\circ t_1(x)=-(1+x)\neq 1-x= t_1\circ r_0(x).$$

A $2\times 2$ representation is:

$$ \begin{align} t_a&\mapsto \begin{pmatrix}1&a\\0&1\end{pmatrix}\\ r_a&\mapsto\begin{pmatrix}-1&a\\0&1\end{pmatrix} \end{align}$$

In you want a discrete lattice, you can pick the subset of the real line, $\mathbb Z$, with the same maps $f_a,r_a$ where $a\in\mathbb Z$, and the same representation.

Answer 2

If one first adds $1$ and then flips in $0$ the point $x$ goes to $x+1$ and winds up at $-x-1.$ But if the flip is done first, the point $x$ goes to $-x$ and winds up at $-x+1.$ So order matters in composing the isometries of the line.