Does the group of permutations of $\omega$ with finite supports interpret arithmetic?

Question

Does the group $G$ of permutation of $\omega$ of finite support interpret $\Bbb N = (\omega, +, \cdot, <, 0, 1)$? I hear about groups that do not interpret bad combinatorial structures all the time, but I am not familiar with groups that do interpret bad structures. The said group $G$ seems bad enough to possibly interpret $\Bbb N$; does it really do so?