This paper presents a biologically inspired fish-robot driven by a single flexible magnetic actuator with a rotating magnetic field in a three-axis Helmholtz coil. Generally, magnetic fish-robots are powered by alternating and gradient magnetic fields, which provide a single motion such as bending the fish-robot's fins. On the other hand, a flexible magnetic actuator driven by an external rotating magnetic field can create several gaits such as the bending vibration, the twisting vibration, and their combination. Most magnetic fish-like micro-robots do not have pectoral fins on the side and are simply propelled by the tail fin. The proposed robot can swim and perform a variety of maneuvers with the addition of pectoral fins and control of the magnetic torque direction. In this paper, we find that the robot's dynamic actuation correlates with the magnetic actuator and the rotating magnetic field. The proposed robot is also equipped with new features, such as a total of six degrees of freedom, a new control method that stabilizes posture, three-dimensional swimming, a new velocity control, and new turning abilities.