Find the character of $\mathbb{C}[S_4/D_8]$.
I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will just be the permutation representation of the cosets.
So I really to need to find a transversal, I can guess that it might be $\{e,(12),(123) \}$ but I cannot see any justification more than a gut feeling. Is there a better way to approach these questions.
If I am not correct in my approach to the original question I would like to know how to compute the cosets.
EDIT: Following on from the comments below.
$$D_8=\langle g,h \mid g^4=h^2=e, hgh^{-1}=g^{-1} \rangle$$
and let us call $g=(1234)$ and $h=(12)$, so $D_8=\langle (1234),(12) \rangle$.