Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$

$K$ is the global number field defined by \( x^{2} - x - 44 \)

$\times$ \(\chi_{ 177 } ( 1, ·)\) \(\chi_{ 177 } ( 176, ·)\)
\(\chi_{ 177 }(1, ·)\) \(\chi_{ 177 } ( 1, ·)\) \(\chi_{ 177 } ( 176, ·)\)
\(\chi_{ 177 }(176, ·)\) \(\chi_{ 177 } ( 176, ·)\) \(\chi_{ 177 } ( 1, ·)\)