Compute the character table for the group $A_{5}$ of even permutations of five elements. You may wish to follow the steps below.

(a) List the conjugacy classes in $A_{5}$ and their orders.

(b) $A_{5}$ acts on $\mathbb{C}^{5}$ by permuting the standard basis vectors. Show that $\mathbb{C}^{5}$ splits as $\mathbb{C} \oplus V$, where $\mathbb{C}$ is the trivial 1-dimensional representation and $V$ is irreducible.

(c) By using the formula for the character of the symmetric square $S^{2} V$,

$$\chi_{S^{2} V}(g)=\frac{1}{2}\left[\chi_{V}(g)^{2}+\chi_{V}\left(g^{2}\right)\right]$$

decompose $S^{2} V$ to produce a 5-dimensional, irreducible representation, and find its character.

(d) Show that the exterior square $\Lambda^{2} V$ decomposes into two distinct irreducibles and compute their characters, to complete the character table of $A_{5}$.

[Hint: You can save yourself some computational effort if you can explain why the automorphism of $A_{5}$, defined by conjugation by a transposition in $S_{5}$, must swap the two summands of $\Lambda^{2} V$.]