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This paper proposes an intelligent reflecting surface (IRS) assisted integrated sensing and communication (ISAC) system operating at the millimeter-wave (mmWave) band. Specifically, the ISAC system combines communication and radar operations and performs, detecting and communicating simultaneously with multiple targets and users. The IRS dynamically controls the amplitude or phase of the radio signal via reflecting elements to reconfigure the radio propagation environment and enhance the transmission rate of the ISAC system. By jointly designing the radar signal covariance (RSC) matrix, the beamforming vector of the communication system, and the IRS phase shift, the ISAC system transmission rate can be improved while matching the desired waveform for radar. The problem is non-convex due to multivariate coupling, and thus we decompose it into two separate subproblems. First, a closed-form solution of the RSC matrix is derived from the desired radar waveform. Next, the quadratic transformation (QT) technique is applied to the subproblem, and then alternating optimization (AO) is employed to determine the communication beamforming vector and the IRS phase shift. For computing the IRS phase shift, we adopt both the majorization minimization (MM) and the manifold optimization (MO). Also, we derive a closed-form solution for the formulated problem, effectively decreasing computational complexity. Furthermore, a trade-off factor is introduced to balance the performance of communication and sensing. Finally, the simulations verify the effectiveness of the algorithm and demonstrate that the IRS can improve the performance of the ISAC system.