Assume that the group $S L_{2}\left(\mathbb{F}_{3}\right)$ of $2 \times 2$ matrices of determinant 1 with entries from the field $\mathbb{F}_{3}$ has presentation

$$\left\langle X, P, Q: X^{3}=P^{4}=1, P^{2}=Q^{2}, P Q P^{-1}=Q^{-1}, X P X^{-1}=Q, X Q X^{-1}=P Q\right\rangle$$

Show that the subgroup generated by $P^{2}$ is central and that the quotient group can be identified with the alternating group $A_{4}$. Assuming further that $S L_{2}\left(\mathbb{F}_{3}\right)$ has seven conjugacy classes find the character table.

Is it true that every irreducible character is induced up from the character of a 1-dimensional representation of some subgroup?

[Hint: You may find it useful to note that $S L_{2}\left(\mathbb{F}_{3}\right)$ may be regarded as a subgroup of $S U_{2}$, providing a faithful 2-dimensional representation; the subgroup generated by $P$ and $Q$ is the quaternion group of order 8 , acting irreducibly.]