R^{-1} {\mathcal{Z}}({\widetilde{\mathcal{G}}},{\mathbb{R}}) R.$$ We now consider in detail the computations and examples for each maximal subgroup of the icosahedral group.

Tetrahedral group ${\mathcal{T}}$
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The tetrahedral group ${\mathcal{T}}$ is the rotational symmetry group of a tetrahedron, generated by a two-fold rotation $g_2$ and a three-fold rotation $g_{3d}$ such that $g_2^2=g^{3}_{3d} = (g_2g_{3d})^3 = e$. It is isomorphic to the alternating group $\mathfrak{A}_4$ and its character table is given by (cf. [@jones]):

where $ \omega = e^{\frac{2\pi i}{3}}$. Note that the representations $E_1$ and $E_2$ are complex, while their direct sum $E := E_1 \oplus E_2$ is real and irreducible in $GL(2,{\mathbb{R}})$. An explicit representation ${\widetilde{\mathcal{T}}}$ of ${\mathcal{T}}$, which is a subgroup of ${\widetilde{\mathcal{I}}}$, is given by This representation can be found with the aid of the software `GAP` [@gap], which is designed for problems in computational group theory (see [@zappa1] for more details). All the subsequent group theoretical computations were performed in `GAP`.

The matrix $R$ as in is such that $$\label{tet_irreps}
\hat{{\mathcal{T}}} :=R^{-1} {\widetilde{\mathcal{T}}}R = \Gamma_1 \oplus \Gamma_2,$$ where $\Gamma_1$ and $\Gamma_2$ are matrix subgroups of $\rho_3$ and $\rho_3'$ as in , respectively, and both are equivalent to the irrep $T$ of ${\mathcal{T}}$. Due to this equivalence, there exists 