Irreducible representation of $S_4$ Could one please point out an irreducible representation of degree 2 of the group $S_4$. Thank you.
 A: Let $\rho^{std}$ be the standard rep. of $S_3$ on $V$ (The dimension of $V$ is equal to $2$). Let $\phi: S_4 \rightarrow S_4/C_2^2$ be a surjective group homomorphism, and $\Omega: S_4/C_2^2 \rightarrow S_3$ a group isomorphism.
Claim: The pullback representation:
\begin{align}
p: S_4 &\rightarrow GL(V)\\
g &\mapsto \rho^{std}(\Omega\circ \phi(g))
\end{align}
is an irreducible rep. of $S_4$ on $V$.
Proof: The homomorphic property of $p$ follows directly from the homomorphic property of $\rho^{std}$, $\phi$ and $\Omega$ therefore $p$ is a representation on  $V$. Let $W$ be a non trivial invarinat subspace of $V$ under $p$:
$$p(g)w = \rho^{std}(\Omega\circ \phi(g)) w = \rho^{std}(h) w \in W$$
Because $\Omega$ and $\phi$ are surjective this holds for all $h \in S_3$. Thus we get the contradiction that $W$ is a non trivial invarinat subspace of $V$ under $\rho^{std}$.
To now prove the assumptions I made we calculate the Quotient group: $S_4/C_2^2$ and we will see that it is isomorphic to $S_3$. After some calculation we find the Elements of $S_4/C_2^2$:
\begin{align}
[C_1] &= \{ (e), (12)(34), (13)(24), (14)(23) \}\\
[C_2] &= \{ (12), (34), (1324), (1423) \}\\
[C_3] &= \{ (13), (24), (1234), (1432) \}\\
[C_4] &= \{ (14), (23), (1243), (1342) \}\\
[C_5] &= \{ (123), (134), (243), (142) \}\\
[C_6] &= \{ (234), (124), (132), (143) \}
\end{align}
the natural map $\phi: S_4 \rightarrow S_4/C_2^2$ is obviously surjective and it is a group homomorphsim as the Quotient group of a Conjugacy class is again a group.
Now we only have to show that $S_4/C_2^2$ is isomorphic to $S_3$. We define the map
\begin{align}
\Omega : S_4/C_2^2 &\rightarrow S_3\\
[C_1] &\mapsto (e)\\
[C_2] &\mapsto (12)\\
[C_3] &\mapsto (13)\\
[C_4] &\mapsto (23)\\
[C_5] &\mapsto (123)\\
[C_6] &\mapsto (132)
\end{align}
which is homomorphic (some more calculations) and obviously bijective thus a group isomorphism. This concludes the proof.
