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

$K$ is the global number field defined by \(x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut -\mathstrut 5 x^{4} \) \(\mathstrut +\mathstrut 4 x^{3} \) \(\mathstrut +\mathstrut 6 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut -\mathstrut 1 \)

$\times$ \(\chi_{ 13 } ( 1, ·)\) \(\chi_{ 13 } ( 3, ·)\) \(\chi_{ 13 } ( 4, ·)\) \(\chi_{ 13 } ( 9, ·)\) \(\chi_{ 13 } ( 10, ·)\) \(\chi_{ 13 } ( 12, ·)\)
\(\chi_{ 13 }(1, ·)\) \(\chi_{ 13 } ( 1, ·)\) \(\chi_{ 13 } ( 3, ·)\) \(\chi_{ 13 } ( 4, ·)\) \(\chi_{ 13 } ( 9, ·)\) \(\chi_{ 13 } ( 10, ·)\) \(\chi_{ 13 } ( 12, ·)\)
\(\chi_{ 13 }(3, ·)\) \(\chi_{ 13 } ( 3, ·)\) \(\chi_{ 13 } ( 9, ·)\) \(\chi_{ 13 } ( 12, ·)\) \(\chi_{ 13 } ( 1, ·)\) \(\chi_{ 13 } ( 4, ·)\) \(\chi_{ 13 } ( 10, ·)\)
\(\chi_{ 13 }(4, ·)\) \(\chi_{ 13 } ( 4, ·)\) \(\chi_{ 13 } ( 12, ·)\) \(\chi_{ 13 } ( 3, ·)\) \(\chi_{ 13 } ( 10, ·)\) \(\chi_{ 13 } ( 1, ·)\) \(\chi_{ 13 } ( 9, ·)\)
\(\chi_{ 13 }(9, ·)\) \(\chi_{ 13 } ( 9, ·)\) \(\chi_{ 13 } ( 1, ·)\) \(\chi_{ 13 } ( 10, ·)\) \(\chi_{ 13 } ( 3, ·)\) \(\chi_{ 13 } ( 12, ·)\) \(\chi_{ 13 } ( 4, ·)\)
\(\chi_{ 13 }(10, ·)\) \(\chi_{ 13 } ( 10, ·)\) \(\chi_{ 13 } ( 4, ·)\) \(\chi_{ 13 } ( 1, ·)\) \(\chi_{ 13 } ( 12, ·)\) \(\chi_{ 13 } ( 9, ·)\) \(\chi_{ 13 } ( 3, ·)\)
\(\chi_{ 13 }(12, ·)\) \(\chi_{ 13 } ( 12, ·)\) \(\chi_{ 13 } ( 10, ·)\) \(\chi_{ 13 } ( 9, ·)\) \(\chi_{ 13 } ( 4, ·)\) \(\chi_{ 13 } ( 3, ·)\) \(\chi_{ 13 } ( 1, ·)\)