  In this paper we introduce a new method for finding Galois groups by
computer. This is particularly effective in the case of Galois groups of
p-extensions ramified at finitely many primes but unramified at the primes
above p. Such Galois groups have been regarded as amongst the most mysterious
objects in number theory. Very little has hitherto been discovered regarding
them despite their importance in studying p-adic Galois representations
unramified at p. The conjectures of Fontaine-Mazur say that they should have no
p-adic analytic quotients (equivalently, the images of the Galois
representations should always be finite), and there are generalizations due to
the first author, suggesting that they should instead have `large' actions on
certain trees.
  The idea is to modify the p-group generation algorithm so that as it goes
along, it uses number-theoretical information to eliminate groups that cannot
arise as suitable quotients of the Galois group under investigation. In the
best cases we obtain a short list of candidates for the Galois group. This
leads to various conjectures.
