Find the group of all rotations (both proper and improper) of a regular tetrahedron.
I understand that the number of elements in the group of proper rotations is 12 but I don't know how to express them in matrix form and I don't know how to find the improper rotation group. Thank you in advance!
OK, then. Let's look at the action of the rotations on the vertices.
First, this is a transitive action; we can clearly send any vertex to any other vertex. If we fix one vertex, what happens to the other three? We rotate that triangle around the symmetry axis (which passes through the fixed vertex), or in the improper case, we reflect across a plane (through the fixed vertex and one of the others). There are three ways each to do that; we can tell which rotation it is or which reflection it is by where one of those vertices is sent.
So, then, that's $4\cdot 3$ rotations and $4\cdot 3$ reflections. Out of the largest possible group $S_4$ - well, that's everything. The full group, including both rotations and reflections, is the symmetric group $S_4$.
What's the rotation group (proper rotations in your terminology)? Well, it's an index-2 subgroup that includes all of the 3-cycles. That has to be $A_4$.