Finite transformations of the one-parumetric relativistic quasiexchange group are found by integrating the nonlinear differential equations that define the infinitesimal transformations of this group. The elements of the quasiexchange group are transformations of the relative momenta $\mathbf{q}$ and $\mathbf{p}$ of three identical relativistic particles that leave invariant the equation $E=\sqrt{\mathbf{p}^2+m^2}+\sqrt{\mathbf{p}^2+4\mathbf{q}^2+4m^2}$ of the energy surface and the element of the three particle phase volume. The group elements are expressed as a function of the parameter $\varphi$ in terms of elliptic Jaeobi functions. In the nonrelativistic ease the latter go over into ordinary trigonometric functions and the finite transformation reduce to a linear representation of the corresponding subgroup of $SO_6$.