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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)