TITLE: What is the order of the symmetry group of the cube?
QUESTION [11 upvotes]: I figured out that there are 24 rotational symmetries, shown below.
Rotation fixing:
faces = 9
diagonals = 8
edges = 6
identity = 1
total = 24
Now, I don't know why the symmetry group of the 3-cube has 48 elements; I know it has to do something with reflection but am unable to picture this. Also, my teacher attached the following hint that I have hard time understanding: "Consider the action of the symmetry group on the set of four diagonals." How is this relevant?

REPLY [6 votes]: I found a nice and elementary way of visualising this. 
Look at linear transformations on $\mathbb{R}^3$. We want the cube to remain fixed.
The fundamental way that is done is by re-labelling the axes $(x, y, z)$. [eg: $(x,y,z)\mapsto (y,z,x)$] You can do that in $3! = 6$ ways.
Thereafter, you can also assign signs to the axes. [eg: $(x,y,z)\mapsto (y,-z,x)$]. You can assign $2$ signs in each position independently $-$ in $2^3 = 8$ ways. 
This exhausts the allowed possibilities. So, the total number of transformations that keep the cube fixed is $3! \times 2^3 = 48$.