  Based on density functional theory and symmetry analysis, we present a
comprehensive investigation of electronic structure, magnetic properties and
possible structural distortion of magnetic kagome metal FeGe. We estimate the
magnetic parameters including Heisenberg and Dzyaloshinskii-Moriya (DM)
interactions, and find that the ferromagnetic nearest-neighbor $J_{1}$
dominates over the others, while the magnetic interactions between nearest
kagome layers favors antiferromagnetic. The N\'{e}el temperature $T_{N}$ and
Curie-Weiss temperature $\theta _{CW}$ are successfully reproduced, and the
calculated magnetic anisotropy energy is also in consistent with the
experiment. However, these reasonable Heisenberg interactions and magnetic
anisotropy cannot explain the double cone magnetic transition, and the DM
interactions, which even exist in the centrosymmetric materials, can result in
this small magnetic cone angle. Unfortunately, due to the crystal symmetry of
the high-temperature structure, the net contribution of DM interactions to
double cone magnetic structure is absent. Based on the experimental $2\times
2\times 2$ supercell, we thus explore the subgroups of the parent phase. Group
theoretical analysis reveals that there are 68 different distortions, and only
four of them (space group $P622$ or $P6_{3}22$) without inversion and mirror
symmetry thus can explain the low-temperature magnetic structure. Furthermore,
we suggest that these four proposed CDW phases can be identified by using Raman
spectroscopy. Since DM interactions are very sensitive to small atomic
displacements and symmetry restrictions, we believe that symmetry analysis is
an effective method to reveal the interplay of delicate structural distortions
and complex magnetic configurations.
