A piston-driven synthetic jet actuator has the potential for application in flow control and fundamental studies of turbulence, although the high-speed flow generated by this actuator is less investigated than a low-speed synthetic jet. The interaction of high-speed jets issued from a piston-driven synthetic jet actuator with multiple orifices is investigated with large eddy simulation (LES). The maximum jet Mach number is related to the maximum pressure inside the actuator regardless of the number of orifices. Temporal variations of the jet Mach number are almost identical for different cycles, and the jet formation in each cycle occurs under the same conditions despite the unsteady nature of the jet interaction. The phase-averaged statistics are used to examine the interaction of the synthetic jets. The converging, merging, and combined regions known for the interaction of continuous jets appear for the interaction of the high-speed synthetic jets slightly before the end of the blowing phase. However, the converging region is not clearly observed at the beginning of the blowing phase because the jets tend to be parallel to each other. Therefore, the combined region forms at a late stage of the blowing phase. Before the jets are combined, velocity fluctuations in the blowing phase become large near the furthest locations where the jets reach. Once the jets merge by their interaction, large velocity fluctuations are observed at the downstream end of the merging region. The probability density functions of velocity fluctuations in the blowing phase tend to deviate from a Gaussian distribution along the centerline of the jets. This deviation is more significant for the two-orifice model than for the four-orifice model under the same actuation frequency.