I have been trying to solve this question and getting nowhere:
The icosahedral group comes equipped with a 3-dimensional representation. What is it? It is irreducible?
Any hints? I know what the representations of $A_5$ look like, but can't do more than that.
There are two distinct, $3$-dimensional, irreducible representations of $A_5$. They have the same image in $\mathrm{GL}_3(\mathbb{R})$ and differ by an outer automorphism of $A_5$.
Both of them correspond to the realization of $A_5$ as the group of symmetries of a icosahedron (or, equivalently, of a dodecahedron): in fact, it is no difficult to see that there are no non-trivial subspaces of $\mathbb{R}^3$ that are invariant for all the symmetries.
For more details, you can have a look at
W. Fulton-J. Harris: Representation Theory, p. 29.