Abstract　 　　
  I’ll talk about the Singular Isothermal Quadrupole (SIQ) lens system, which is the simplest lens model that can produce four images. The basic properties of the SIQ lens are studied, including deflection potential, deflection angle, magnification, critical curve, caustic, pseudo-caustic and transition locus. With this simple lens, we study the situations that a point source infinitely approaches a cusp or a fold. Unlike the traditional views, the sums of magnifications of cusp image triplet or the fold image pair are usually not equal to 0. Then, I’ll talk about the caustics that are composed of two overlapping layers. When a point source traverses such overlapping caustics, the image numbers change by 4. We also focus on the magnification patterns of the quadrupole lenses perturbed by m=3, 4 and 5 modes, and found that one, two, and three butterfly or swallowtail singularities can be produced respectively. With the increasing intensities of the high-order perturbations, the singularities grow up to bring sixfold image regions. If these perturbations are large enough to let two or three of the butterflies or swallowtails contact, eightfold or tenfold image regions can be produced as well.