Intelligent reflecting surface (IRS) has recently been envisioned to offer unprecedented massive multiple-input multiple-output (MIMO)-like gains by deploying large-scale and low-cost passive reflection elements. By adjusting the reflection coefficients, the IRS can change the phase shifts on the impinging electromagnetic waves so that it can smartly reconfigure the signal propagation environment and enhance the power of the desired received signal or suppress the interference signal. In this paper, we consider downlink multigroup multicast communication systems assisted by an IRS. We aim for maximizing the sum rate of all the multicasting groups by the joint optimization of the precoding matrix at the base station (BS) and the reflection coefficients at the IRS under both the power and unit-modulus constraint. To tackle this non-convex problem, we propose two efficient algorithms under the majorization--minimization (MM) algorithm framework. Specifically, a concave lower bound surrogate objective function of each user's rate has been derived firstly, based on which two sets of variables can be updated alternately by solving two corresponding second-order cone programming (SOCP) problems. Then, in order to reduce the computational complexity, we derive another concave lower bound function of each group's rate for each set of variables at every iteration, and obtain the closed-form solutions under these loose surrogate objective functions. Finally, the simulation results demonstrate the benefits in terms of the spectral and energy efficiency of the introduced IRS and the effectiveness in terms of the convergence and complexity of our proposed algorithms.