Let G_n(Lambda) be the semi-direct product of the symmetric group S_n by the Steinberg group St_n(Lambda) of a ring Lambda. We first prove that G_n(Lambda) has a Coxeter-type presentation. The canonical morphism St_n(Lambda) to GL_n(Lambda) extends to a group homomorphism Phi: G_n(Lambda) to GL_n(Lambda). We next determine the kernel of Phi in the stable case (n=infinity). We also give an expression for the generator of the algebraic K-group K_2(Z) of the integers in terms of permutation matrices.