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

$K$ is the global number field defined by \(x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut +\mathstrut 2 x^{4} \) \(\mathstrut +\mathstrut 8 x^{3} \) \(\mathstrut -\mathstrut x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 7 \)

$\times$ \(\chi_{ 19 } ( 1, ·)\) \(\chi_{ 19 } ( 18, ·)\) \(\chi_{ 19 } ( 7, ·)\) \(\chi_{ 19 } ( 8, ·)\) \(\chi_{ 19 } ( 11, ·)\) \(\chi_{ 19 } ( 12, ·)\)
\(\chi_{ 19 }(1, ·)\) \(\chi_{ 19 } ( 1, ·)\) \(\chi_{ 19 } ( 18, ·)\) \(\chi_{ 19 } ( 7, ·)\) \(\chi_{ 19 } ( 8, ·)\) \(\chi_{ 19 } ( 11, ·)\) \(\chi_{ 19 } ( 12, ·)\)
\(\chi_{ 19 }(18, ·)\) \(\chi_{ 19 } ( 18, ·)\) \(\chi_{ 19 } ( 1, ·)\) \(\chi_{ 19 } ( 12, ·)\) \(\chi_{ 19 } ( 11, ·)\) \(\chi_{ 19 } ( 8, ·)\) \(\chi_{ 19 } ( 7, ·)\)
\(\chi_{ 19 }(7, ·)\) \(\chi_{ 19 } ( 7, ·)\) \(\chi_{ 19 } ( 12, ·)\) \(\chi_{ 19 } ( 11, ·)\) \(\chi_{ 19 } ( 18, ·)\) \(\chi_{ 19 } ( 1, ·)\) \(\chi_{ 19 } ( 8, ·)\)
\(\chi_{ 19 }(8, ·)\) \(\chi_{ 19 } ( 8, ·)\) \(\chi_{ 19 } ( 11, ·)\) \(\chi_{ 19 } ( 18, ·)\) \(\chi_{ 19 } ( 7, ·)\) \(\chi_{ 19 } ( 12, ·)\) \(\chi_{ 19 } ( 1, ·)\)
\(\chi_{ 19 }(11, ·)\) \(\chi_{ 19 } ( 11, ·)\) \(\chi_{ 19 } ( 8, ·)\) \(\chi_{ 19 } ( 1, ·)\) \(\chi_{ 19 } ( 12, ·)\) \(\chi_{ 19 } ( 7, ·)\) \(\chi_{ 19 } ( 18, ·)\)
\(\chi_{ 19 }(12, ·)\) \(\chi_{ 19 } ( 12, ·)\) \(\chi_{ 19 } ( 7, ·)\) \(\chi_{ 19 } ( 8, ·)\) \(\chi_{ 19 } ( 1, ·)\) \(\chi_{ 19 } ( 18, ·)\) \(\chi_{ 19 } ( 11, ·)\)