Intelligent reflecting surface (IRS) has recently attracted increasing research interests due to its great potential in enhancing the energy and spectrum efficiency for wireless networks. In this paper, we shall investigate the downlink transmit power optimization problem for an IRS-assisted multigroup multicast system, where the beamformers at the base station (BS) and the phase shifts at the IRS are jointly optimized. We take into account the quality-of-service (QoS) requirements of all users in each multicasting group as well as the constant-envelope reflection of all IRS elements. The formulated problem turns out to be a non-convex quadratically constrained bi-quadratic programming problem, due to the intricate coupling between the optimization variables and the non-convex constant-envelope constraints. To this end, we present the alternating optimization method with matrix lifting to decouple the optimization variables, followed by ensuring the feasibility of the rank-one constraints via introducing the difference-of-convex (DC) function representation. We develop an effective alternating algorithm to solve the joint optimization problem. Extensive simulation results show the superiority of the proposed algorithm in reducing the transmit power of multigroup multicast systems.