  We prove that the quotients of the group algebra of the braid group on 3
strands by a generic quartic and quintic relation respectively, have finite
rank. This is a special case of a conjecture by Brou\'{e}, Malle and Rouquier
for the generic Hecke algebra of an arbitrary complex reflection group.
Exploring the consequences of this case, we will prove that we can determine
completely the irreducible representations of this braid group for dimension at
most 5, thus reproving a classification of Tuba and Wenzl in a more general
framework.
