What is a permutation representation in regard to group actions

I have read the definition of a permutation representation from Dummit and Foote, and Wiki, but I don't understand.

Can I please have an example? I get the impression that we can write a group action normally as:

$$G\times X\to X, (g,x)\mapsto g\cdot x$$

Or we can write it as a permutation representation, i.e. a group action does:

$$\begin{pmatrix}1&2&3&4\\\sigma(1)&\sigma(2)&\sigma(3)&\sigma(4)\end{pmatrix}$$

Is that all a permutation representation refers to?

• Here is an example. The group of rotations ($\cong Z_3$) acts on the set $V$ of vertices (label them $1,2,3$) of an equilateral triangle -- one would often just say that $Z_3$ acts on $V$ by rotations. Then we get a mapping $\pi$ from $Z_3$ to $S(V)$, which is just $S_3$ using the labels. $\pi$ takes the $0^\circ$ rotation to $e$ (it fixes all vertices) the $120^\circ$ rotation to the permutation $(1 \, 2 \, 3)$, and the $240^\circ$ rotation to $(1 \, 3 \, 2)$. $\pi$ is a group homomorphism -- it's easy to check this here, but by the way group actions are defined you always get a homomorphism. – John Brevik Aug 12 '15 at 14:28
• Thanks @JohnBrevik that makes sense, I appreciate your comment. – Galois in the Field Aug 14 '15 at 5:23

A permutation representation is the same thing as a group action, basically. If a group $G$ acts on a set $X$, then the action gives a homomorphism from $G$ to the group of permutations on $X$, which is the definition of a permutation representation of $G$ on $X$.