We have intensively investigated the antiferromagnetic phase transition in the ternary rare-earth metal silicide Er 5Ir 4Si 10 single crystal by performing the high-resolution measurement of the low-temperature specific heat under zero-magnetic field and the AC magnetization. In the temperature dependence of important physical quantities associated with the antiferromagnetic phase transition, we have observed anomalies with the antiferromagnetic long-range ordering at Neel temperature T N. We have confirmed that T N is 3.5 K. In addition, we have first observed two surprising results. Firstly, a shoulder was observed in the vicinity of 2 K in addition to the sharp peak at T N corresponding to the antiferromagnetic phase transition in the high-resolution measurement of the low-temperature specific heat. Secondly, the anomaly of the AC magnetization at T N depends to the magnetic field direction. We have clearly observed the anomaly of the AC magnetization at T N when the AC magnetic field orientation is parallel to the c-axis, whereas we have observed no anomaly of the AC magnetization at T N when the AC magnetic field orientation is perpendicular to the c-axis. These results clarify that our Er 5Ir 4Si 10 single crystal is a quasi-two-dimensional antiferromagnet and has no three-dimensional magnetic structure of the Er 3+ local moments. However, we have observed a peak of the AC magnetization around 2 K when the AC magnetic field orientation is perpendicular to the c-axis. This temperature corresponds to that at which a shoulder is observed in the measurement of the low-temperature specific heat. Furthermore, we have observed no frequency dependence of the AC magnetization which is ordinarily observed in the spin glass state. This result means that there is no disorder in our Er 5Ir 4Si 10 single crystal because the crystal structure of Er 5Ir 4Si 10 has the tetragonal crystal structure in which the octagons of Er 3+ ions are stacked. In addition, both tetragons and octagons of Er 3+ local moments have no magnetic frustration. At last we can conclude that both the shoulder of the low-temperature specific heat in the vicinity of 2 K and the peak of the AC magnetization around 2 K, which is only observed when the AC magnetic field orientation is perpendicular to the c-axis, correspond to the crystalline-electric-effect in the plane which is perpendicular to the c-axis of the tetragonal crystal structure.