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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 $D_8=\langle (1234),(12) \rangle$.

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  • $\begingroup$ You can't compute the cosets until you have specified the subgroup, and you haven't done that yet! Just referring to it as $D_8$ doesn' t define it unambiguously, because $S_4$ has three conjugate subgroups isomorphic to $D_8$. $\endgroup$ – Derek Holt May 30 '15 at 15:36
  • $\begingroup$ Isnt the subgroup just $D_8=\langle g,h \mid g^4=h^2=e, hgh^{-1}=g^{-1} \rangle$? $\endgroup$ – Permian May 30 '15 at 16:45
  • $\begingroup$ But you have to say what $g$ and $h$ are as elements of $S_4$. There is more than one way of doint that, and it doesn't matter which you choose, but you cannot find a transversal unitl you have done that. $\endgroup$ – Derek Holt May 30 '15 at 22:00
  • $\begingroup$ As it happens, whatever choice you make for the subgroup $D_8$ of $S_4$ (there are three possible choices), you could choose $\{ e,(1,2,3),(1,3,2)\}$ as the tranversal, but you would understand the calculation better if you knew what $H$ was. $\endgroup$ – Derek Holt May 30 '15 at 22:05
  • $\begingroup$ Ok I see this now and have edited the quesiton $\endgroup$ – Permian May 31 '15 at 10:48

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