Showing that a subgroup of Sym(4) is isomorphic to D_8

Homework Statement

Let $R$ be the set of all polynomials with integer coefficients in the independent variables $x_1, x_2, x_3, x_4$.
$S_4$ acts on $R$ by the group action $\sigma \cdot p(x_1,x_2,x_3,x_4) = p(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)},x_{\sigma(4)})$. Exhibit all permutations in $S_4$ that stabilize the element $x_1x_2 + x_3x_4$ and prove that they form a subgroup isomorphic to the dihedral group of order 8.

The Attempt at a Solution

So, by listing the elements of $S_4$ and seeing how they act on $x_1x_2+x_3x_4$, I found that the stabilizer of $x_1x_2+x_3x_4$ is $\{1,(12),(34),(12)(34),(13)(24),(14)(23),(1324),(1423) \}$.

My problem is showing that this is isomorphic to the dihedral group of order 8. I think the correspondence is clear if we label the vertices of a square and consider the permutations as symmetries of the square, but I feel that exhibiting this 1-1 correspondence in that much detail is too computationally intensive and that there should be an easier way.

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Mentor
So, by listing the elements of $S_4$ and seeing how they act on $x_1x_2+x_3x_4$, I found that the stabilizer of $x_1x_2+x_3x_4$ is $\{1,(12),(34),(12)(34),(13)(24),(14)(23),(1324),(1423) \}$.

My problem is showing that this is isomorphic to the dihedral group of order 8. I think the correspondence is clear if we label the vertices of a square and consider the permutations as symmetries of the square, but I feel that exhibiting this 1-1 correspondence in that much detail is too computationally intensive and that there should be an easier way.
What is $D_8\,$? And no, I'm not asking about the index, nor do I want to know that it is the dihedral group of order eight. Without this information, I'd say pin a screenshot of this Wiki page and you're done: https://de.wikipedia.org/wiki/Diedergruppe#Permutations-Darstellung
Just observe, that you have an odd numbering of the square: $(12)$ and $(34)$ are the diagonals and not neighboring points.

Mr Davis 97