  Between 1994 and 1998, the work of M. Brou\'e, G. Malle, and R. Rouquier
generalized in a natural way the definition of the Hecke algebra associated to
a finite Coxeter group, for the case of an arbitrary complex reflection group.
Attempting to also generalize the properties of the Coxeter case, they stated a
number of conjectures concerning these Hecke algebras. One specific example of
importance regarding those yet unsolved conjectures is the so-called BMR
freeness conjecture. This conjecture is known to be true apart from 16 cases,
that are almost all the exceptional groups of rank 2. These exceptional groups
of rank 2 fall into three families: the tetrahedral, octahedral and icosahedral
family. We prove the validity of the BMR freeness conjecture for the
exceptional groups belonging to the first two families, using a case-by-case
analysis and we give a nice description of the basis, similar to the classical
case of the finite Coxeter groups. We also give a new consequence of this
conjecture, by obtaining the classification of irreducible representations of
the braid group on 3 strands in dimension at most 5, recovering results of Tuba
and Wenzl.
