The Anderson transitions in a random magnetic field in three dimensions are investigated in detail by means of the transfer-matrix method with high accuracy. Both systems with and systems without an additional random scalar potential are considered. We find the critical exponent for the localization length to be with a strong random scalar potential. Without it, the exponent is smaller but increases with the system size and extrapolates to the above value within the error bars. These results support the conventional classification of universality classes according to symmetry. The mobility edge trajectory in the random magnetic field is also obtained.