A crystallographic group in $n$ dimensions is basically a discrete subgroup of the Euclidean group $E(n)$ (see ncatlab).
Is there anything known about irreducible representations of these groups? I would guess maybe it is related to the study of irreps of the corresponding point group but I am not sure.
For example, in $2$-dimensions, these are called the wallpaper groups and there are 17 of them. Is there a reference on the irreducible representations of these groups? Are any of the irreps finite dimensional?
I'll stick to the notation from the ncatlab article linked in the post. $S$ is the crystallographic group, $N \cong \mathbb{Z}^n$ is the normal subgroup of translations, and $G = S/N$ is the point group (a finite group). I'll also assume you are interested in finite dimensional representations.
Note that $G$ acts on $N$ by conjugation, and therefore also acts on the set $N^ \vee$ of irreducible representations (i.e. characters) of $N$. If we take an irreducible representation of $S$ and restrict it to $N$ then necessarily it decomposes as a sum of characters living on a single $G$ orbit of this action (if not, the sum of the subspaces on a single orbit would form a subrepresentation).
So now for each $G$ orbit on $N^\vee$ we can ask what $S$-irreducibles restrict to things on it? The answer to this depends on the orbit. Let's consider the two extreme cases:
If we take the trivial character of $N$, this is a $G$-orbit all on it's own. An representation of $S$ that restricts to the trivial representation on $N$ is just a representation of $S/N = G$. So on this orbit we just get the irreducible representations of $G$.
On the other extreme, if we pick a character $\chi$ such that $G$ acts freely on the $G$-orbit of $\chi$ then we see that $Ind_N^S(\chi)$ is irreducible as every character in the orbit just appears once. So we only get a single $S$-irreducible on this orbit.
The general case more complicated to describe completely so I think I won't do that right now, but is in between these two cases. If $\chi$ is a character of $N$ then there is some subgroup $H \subset G$ that stabilizes $\chi$ and we also need to account for how $H$ acts on the $\chi$-isotypic component.