MSG 164.90

\[\begin{align*} &E_{2}^{1,-1}=\mathbb{Z}_2^2\\ &E_{1}^{0,-1}=\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{A}_5}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{C}_3}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{D}_1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{D}_3}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{D}_5}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{E}_1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{E}_3}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{F}_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{F}_2}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{F}_3}]\\ &E_{1}^{1,-1}=\mathbb{Z}[\boldsymbol{b}^{(1)}_{a_1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(1)}_{a_3}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{b_1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(1)}_{b_2}]\oplus\mathbb{Z}[\boldsymbol{b}^{(1)}_{c_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{e_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{e_2}]\oplus\mathbb{Z}[\boldsymbol{b}^{(1)}_{f_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{h_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{h_2}]\\ &E_{1}^{2,-1}=2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\alpha _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\beta _1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(2)}_{\gamma _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\delta _1}] \end{align*}\]

\[\begin{align*} &[X^{(1)}]^{-1}=\left( \begin{array}{cccccccccc} a_1 & a_3 & b_1 & b_2 & c_1 & e_1 & e_2 & f_1 & h_1 & h_2 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ -1 & -1 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & -1 & -1 & 0 & 0 & 0 & 1 & 1 \\ \end{array} \right)\\ &[V^{(0)}]^{-1}=\left( \begin{array}{cccccccccc} \text{A}_5 & \text{C}_3 & \text{D}_1 & \text{D}_3 & \text{D}_5 & \text{E}_1 & \text{E}_3 & \text{F}_1 & \text{F}_2 & \text{F}_3 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & -2 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & -2 & 0 & 1 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\ &\Lambda^{(0)}=\left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ \end{array} \right)\\ &[V^{(1)}]^{-1}=\left( \begin{array}{cccccccccc} a_1 & a_3 & b_1 & b_2 & c_1 & e_1 & e_2 & f_1 & h_1 & h_2 \\ 1 & 1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & -1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)\\ &\Sigma^{(1)}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right) \end{align*} \]