We report on spherical neutron polarimetry and unpolarized neutron diffraction in zero magnetic field as well as flipping ratio and static magnetization measurements in high magnetic fields on the multiferroic square lattice antiferromagnet Ba$_2$CoGe$_2$O$_7$. We found that in zero magnetic field the magnetic space group is $Cmm2$ with sublattice magnetization parallel to the [100] axis of this orthorhombic setting. The spin canting has been found to be smaller than $0.2^circ$ in the ground state. This assignment is in agreement with the field-induced changes of the magnetic domain structure below 40 mT as resolved by spherical neutron polarimetry. The magnitude of the ordered moment has been precisely determined. Above the magnetic ordering temperature short-range magnetic fluctuations are observed. Based on the high-field magnetization data, we refined the parameters of the recently proposed microscopic spin model describing the multiferroic phase of Ba$_2$CoGe$_2$O$_7$.
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