  We give a new infinite family of group homomorphisms from the braid group B_k
to the symmetric group S_{mk} for all k and m \geq 2. Most known permutation
representations of braids are included in this family. We prove that the
homomorphisms in this family are non-cyclic and transitive. For any divisor l
of m, 1\leq l < m, we prove in particular that if \frac{m}{l} is odd then there
are 1 + \frac{m}{l} non-conjugate homomorphisms included in our family.
  We define a certain natural restriction on homomorphisms B_k to S_n, common
to all homomorphisms in our family, which we term 'good', and of which there
are two types.
  We prove that all good homomorphisms B_k to S_{mk} of type 1 are included in
the infinite family of homomorphisms we gave. For m=3, we prove that all good
homomorphisms B_k to S_{3k} of type 2 are also included in this family.
  Finally, we refute a conjecture made by Matei and Suciu regarding permutation
representations of braids and give an updated conjecture.
