icosahedral group ${\mathcal{I}}$, namely the tetrahedral group ${\mathcal{T}}$ or the dihedral groups ${\mathcal{D}}_{10}$ and ${\mathcal{D}}_6$, and let $\widetilde{{\mathcal{G}}}$ be a crystallographic representation of ${\mathcal{G}}$ embedded into the hyperoctahedral group $B_6$. Without loss of generality, we consider $\widetilde{{\mathcal{G}}}$ as a subgroup of the crystallographic representation $\widetilde{{\mathcal{I}}}$ given in ; let ${\mathcal{C}}_{B_6}({\widetilde{\mathcal{G}}})$ denote the conjugacy class of ${\widetilde{\mathcal{G}}}$ in $B_6$. We have the following (for the proof, see [@zappa1]):

\[reps\] Let ${\mathcal{G}}$ be a maximal subgroup of ${\mathcal{I}}$. Then for every $P \in {\mathcal{C}}_{B_6}({\widetilde{\mathcal{G}}})$ there exist *exactly* two crystallographic representations of ${\mathcal{I}}$, ${\mathcal{H}}_1, {\mathcal{H}}_2 \in {\mathcal{C}}_{B_6}(\widetilde{{\mathcal{I}}})$, such that $P = {\mathcal{H}}_1 
 \cap {\mathcal{H}}_2$.

We point out that in the case of achiral icosahedral symmetry, the symmetry group to be considered is the Coxeter group $H_3$ [@coxeter]. Due to the isomorphism $H_3 \simeq {\mathcal{I}}\times {\mathbb{Z}}_2$, the representation theory of $H_3$ easily follows from this direct product structure. In particular, the representation ${\mathcal{H}}_3 := {\widetilde{\mathcal{I}}}\otimes \Gamma$, where $\Gamma = \{1, -1\}$ is the non-trivial irreducible representation of ${\mathbb{Z}}_2$, is a six-dimensional crystallographic representation of $H_3$ in the sense of Definition \[cryst\_def\]. Therefore, all the analysis developed in the following still holds for achiral model sets and arrays.

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   Generator         Irrep $\rho_3 \simeq T_1$                Irrep 