What is a permutation representation in regard to group actions

Question

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?

Answer 1

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 $X$.