  High-resolution time-of-flight powder neutron diffraction and high-field
magnetization were measured to investigate the magnetic structure and existence
of a field-induced magnetic phase transition in the distorted kagome
antiferromagnet Cs$_2$Cu$_3$SnF$_{12}$. Upon cooling from room temperature, the
compound undergoes a structural phase transition at $T_\textrm{t}=185$ K from
the rhombohedral space group $R\bar{3}m$ with the perfect kagome spin network
to the monoclinic space group $P2_1/n$ with the distorted kagome planes. The
distortion results in three inequivalent exchange interactions among the
$S=1/2$ Cu$^{2+}$ spins that magnetically order below $T_\textrm{N}=20.2$ K.
Magnetization measured with a magnetic field applied within the kagome plane
reveals small in-plane ferromagnetism resulting from spin canting. On the other
hand, the out-of-plane magnetization does not show a clear hysteresis loop of
the ferromagnetic component nor a prominent anomaly up to 170 T, with the
exception of the subtle knee-like bend around 90 T, which could indicate the
1/3 magnetization plateau. The combined analysis using the irreducible
representations of the magnetic space groups and magnetic structure refinement
on the neutron powder diffraction data suggests that the magnetic moments order
in the magnetic space group $P2_1'/n'$ with the all-in-all-out spin structure,
which by symmetry allows for the in-plane canting, consistent with the in-plane
ferromagnetism observed in the magnetization.
