  \]

  where each \( \phi_i \) is given by

  \begin{align*}
    \phi_1 &amp;:
    a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6} \mapsto
    a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}, \\

    \phi_2 &amp;:
    a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6} \mapsto
    a - b \sqrt{2} + c \sqrt{3} - d \sqrt{6}, \\

    \phi_3 &amp;:
    a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6} \mapsto
    a + b \sqrt{2} - c \sqrt{3} - d \sqrt{6}, \\

    \phi_4 &amp;:
    a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6} \mapsto
    a - b \sqrt{