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Exact expressions for the radiation quality factor Q and the gain pattern function of the corresponding antenna are derived and their properties are analyzed for crossed electric and magnetic dipole moment configuration. Both omni-directional and directional antennas with linear, elliptical, or circular polarization can achieve the same minimum radiation Q. The maximum gain of the antenna ranges from 3/2 to 3. The maximum gain of three can be achieved only for linear polarization when the source vectors make a right angle and the amplitudes of the TM and the TE modes are equal. The increase in the maximum gain from 3/2 for omni-directional antennas to 3 for directional antennas does not require any increase in the radiation Q, which suggests that practical directional antennas with higher gain may be realized with smaller size compared with small omni-directional antennas assuming similar operation bandwidths.