$\widetilde{G}$ stabilises a lattice ${\mathcal{L}}$ in ${\mathbb{R}}^d$, with $d > k$, i.e. $\widetilde{G}$ is a subgroup of the point group ${\mathcal{P}}$ of ${\mathcal{L}}$;

2.  $\widetilde{G}$ is reducible in $GL(d,{\mathbb{R}})$ and contains an irreducible representation $\rho_k$ of $G$ of degree $k$, i.e. $$\label{cryst}  
    \widetilde{G} \simeq \rho_k \oplus \rho', \qquad \deg(\rho') = d-k.$$

The condition ($\mathbf{C2}$) is necessary for the construction of quasicrystals with $G$-symmetry via the cut-and-project method [@grimm; @senechal].

Chiral icosahedral symmetry is described by the icosahedral group ${\mathcal{I}}$, which corresponds to the set of all the rotations that leave an icosahedron invariant. It has order $60$ and is the largest finite subgroup of the special orthogonal group $SO(3)$ [@artin]. It is isomorphic to the alternating group $\mathfrak{A}_5$, and has presentation $${\mathcal{I}}= \langle g_2, g_3 : g_2^2=g_3^3=(g_2g_3)^5=e \rangle,$$ where $g_2$ and $g_3$ represent, geometrically, a two- and a three-fold rotation, respectively. The element $g_5:=g_2g_3$ is a five-fold rotation, hence ${\mathcal{I}}$ is non-crystallographic in ${\mathbb{R}}^3$. Its character table is the following ($\tau := \frac{1}{2} \left(1+\sqrt{5}\right)$ denotes the golden ratio, and $\tau' := 1-\tau$ its Galois conjugate):

The minimal crystallographic dimension of ${\mathcal{I}}$ is six (cf. Definition \[cryst\_def\]) [@levitov]; an explicit crystallographic representation ${\widetilde{\mathcal{I}}}$ of ${\mathcal{I}}$ is given by [@zappa1]: There are 