Sage's built-in help is useful here.

```
sage: PermutationGroup?
Definition: PermutationGroup(gens=None, gap_group=None, domain=None, canonicalize=True, category=None)
Docstring:
Return the permutation group associated to x (typically a list of
generators).
INPUT:
* "gens" - list of generators (default: "None")
* "gap_group" - a gap permutation group (default: "None")
* "canonicalize" - bool (default: "True"); if "True", sort
generators and remove duplicates
OUTPUT:
* A permutation group.
```

So indeed you get a permutation group, with these things as generators. You can't multiply groups.

A similar look at help will show you that `Permutation`

does not give an element of a group, though you can convert an element of such a group to one.

But you *can* get group elements from your group. Try this.

```
sage: G = SymmetricGroup(4)
sage: G([(1,2)])
(1,2)
sage: G([(1,2)])*G([(1,3)])*G([(1,4)])
(1,2,3,4)
```