Assume $G$ is a finite group.
I am trying to construct the character table of $Q_8$, which is defined by
$$Q_8=\{\pm 1,\pm i, \pm j,\pm k \}, \ i^2=j^2=k^2=-1, \ ij=k,jk=i,ki=j$$
By considering the $G'$, I can prove that $|G|=8=1^2+1^2+1^2+1^2+2^2$. Therefore $G$ has four $1$-dimensional representations and one $2$-dimensional representation. So
$$ \begin{array}{|c|c|c|c|} \hline &1& -1 & \{\pm i \} & \{\pm j \} & \{\pm k \} \\ \hline \chi_0 & 1 & 1 & 1 & 1 & 1 \\ \hline \chi_1 & 1 & ? & ?&? & ? \\ \hline \chi_2 & 1 & ? &? & ?& ? \\ \hline \chi_3 & 1 & ? & ?&? & ? \\ \hline \chi_4 & 2 & ? & ? & ? & ? \\ \hline \end{array} $$
I guessing the first vertical column is a 2 at the bottom because $tr(I_2)=2$. The first row is $1$'s because it is the 1-d trivial representation.
I am unsure of how to proceed from here. Should I be able to identity the $\chi_i$'s?
Please do not give full solutions as otherwise I will not learn.
If it helps its fairly easy to show that $Q_8 / Q_8' \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$