  Due to the lack of full rotational symmetry in condensed matter physics,
solids exhibit new excitations beyond Dirac and Weyl fermions, of which the
six-fold excitations have attracted considerable interest owing to the presence
of the maximum degeneracy in bosonic systems. Here, we propose that a single
linear dispersive six-fold excitation can be found in the electride
Li$_{12}$Mg$_3$Si$_4$ and its derivatives. The six-fold excitation is formed by
the floating bands of elementary band representation -- $A@12a$ -- originating
from the excess electrons centered at the vacancies (${\it i.e.}$, the $12a$
Wyckoff sites). There exists a unique topological bulk-surface-edge
correspondence for the spinless six-fold excitation, resulting in trivial
surface 'Fermi arcs' but nontrivial hinge arcs. All energetically-gapped
$k_z$-slices belong to a two-dimensional (2D) higher-order topological
insulating phase, which is protected by a combined symmetry ${\mathcal
T}{\widetilde S_{4z}}$ and characterized by a quantized fractional corner
charge $Q_{corner}=\frac{3|e|}{4}$. Consequently, the hinge arcs are obtained
in the hinge spectra of the $\widetilde S_{4z}$-symmetric rod structure. The
state with a single six-fold excitation, stabilized by both nonsymmorphic
crystalline symmetries and time-reversal symmetry, is located at the phase
boundary and can be driven into various topologically distinct phases by
explicit breaking of symmetries, making these electrides promising platforms
for the systematic studies of different topological phases.
