I found a footnote on page 140 of this book that reads:
For a general $d$-dimensional quasicrystal, there is more than one $d$-dimensional irreducible representation. For the case of two-dimensional pentagonal symmetry, which describes the original Penrose tilings, there are two distinct two-dimensional representations of the pentagonal group.
From which my take-home message is: for regular crystals in $d$ dimensions, there is only one $d$ dimensional irreducible representation.
But then why does the dihedral group of order $n$ (even, for instance) have $(n-2)/2$ two-dimensional irreducible representations, as listed here?