  The Galois/monodromy group of a family of geometric problems or equations is
a subtle invariant that encodes the structure of the solutions. Computing
monodromy permutations using numerical algebraic geometry gives information
about the group, but can only determine it when it is the full symmetric group.
We give numerical methods to compute the Galois group and study it when it is
not the full symmetric group. One algorithm computes generators while the other
gives information on its structure as a permutation group. We illustrate these
algorithms with examples using a Macaulay2 package we are developing that
relies upon Bertini to perform monodromy computations.
