The question I need to answer is in the title. I'm learning group actions now, and I am understanding that the action of an element of a group acting on a set produces a permutation of that set. I am also understanding that an equivalent notion of group action is a homomorphism from the group onto the a symmetric group associated with the group. In this case, the set acted upon is $D_8$, the dihedral group of order 8, the same set as the group.
I like using the group product table, where we can easily see that, taking one element from $D_8$, and multiplying by every other element, we produce a permutation of the set. For example, $A=D_8=\{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$, take the element $r$ from $D_8$, then $rA=\{r,r^2,r^3,1,sr^3,sr^2,sr,s\}$. And its pretty cool that using every element from $D_8$ gives a distinct permutation of the set.
My question is I am not really sure what this question means (the one in the title)? I assume we are talking about a homomorphism: $$\phi:D_8 \to \mathbb{S}_8$$ And I guess If I knew exactly how to write the map of the homomorphism I could determine the image, but here is where I am stuck. How do I visualize the map of this homomorphism from $D_8$ to $\mathbb{S}_8$?