Permutation representation of rotations of a cube In my lecture notes, an example is as follows:
Let $G$ be the group of rotations of the cube and let $X$ be the set of faces. We can think of an element of the permutation representation as being a way of writing a complex number on each face.
I’m struggling to see how this works, could anyone possibly help break this down and help me understand a bit better?
 A: If $G$ is a group acting on a set $X$ then $G$ acts naturally on the vector space of functions $f:  X \to \mathbb{C}$.   The action is given by $(g \cdot f)(x) = f(gx)$.  We see that $(g \cdot f)$ is a new function from $X$ to $\mathbb{C}$, which permutes which elements of $x$ get assigned the different values of the function.
In this case, $G$ is the rotations of the cube and $X$ is the set of faces of the cube.  "Writing a complex number on each face"  is really the same thing as a function from $X$ to $\mathbb{C}$, the permutation representation is the action of $G$ on the collection of all such functions.   In this case the representation is $6$ dimensional, because there are 6 faces and we get to pick a value of the function for each face.
Let's work out an explicit example of the action in this case.  Suppose $f$ is the function:
$$f(top) = 2 \ \ \ f(bottom) = 7 \ \ \ \ f(front) = i$$
$$f(back) = \pi \ \ \ f(right) = 3+i \ \ \ f(left) = -47$$
And now let's take $g \in G$ to be the element where we rotate the cube 90 degrees around the axis through the middle of the top and bottom faces.  The new function $g \cdot f$ we get is:
$$(g \cdot f)(top) = 2 \ \ \ (g \cdot f)(bottom) = 7 \ \ \ \ (g \cdot f)(front) = 3+i$$
$$(g \cdot f)(back) = -47 \ \ \ (g \cdot f)(right) = \pi \ \ \ (g \cdot f)(left) = i$$
So if you like, we have just permuted the labels on the faces.
