  An algebraic characterization of the contractions of the Poincar\'e group
permits a proper construction of a non-relativistic limit of its tachyonic
representation. We arrive at a consistent, nonstandard representation of the
Galilei group which was disregarded long ago by supposedly unphysical
properties. The corresponding quantum (and classical) theory shares with the
relativistic one their fundamentals, and serves as a toy model to better
comprehend the unusual behavior of the tachyonic representation. For instance,
we see that evolution takes place in a spatial coordinate rather than time, as
for relativistic tachyons, but the modulus of the three-momentum is the same
for all Galilean observers, leading to a new dispersion relation for a Galilean
system. Furthermore, the tachyonic objects described by the new representation
cannot be regarded as localizable in the standard sense.
