 Solution

Use the given square-class assignments and ${\omega }^3=1$, $1+\omega +{\omega }^2=0$. The character values, in the specified class order, areFor example the fourth entry of ${\chi }_S$ is $((-{\omega }^2{)}^2-\omega )/2=0$, and that of ${\chi }_A$ is $\omega$. A class of centralizer order $c$ has size $24/c$, so a character norm is the sum of its squared moduli divided by these centralizer orders. ThereforeBoth are genuine characters by their symmetric/exterior-square construction. A character norm is the sum of squares of its irreducible multiplicities, so both characters are irreducible.