  In analogy to the disjoint cycle decomposition in permutation groups, Ore and
Specht define a decomposition of elements of the full monomial group and
exploit this to describe conjugacy classes and centralisers of elements in the
full monomial group. We generalise their results to wreath products whose base
group need not be finite and whose top group acts faithfully on a finite set.
We parameterise conjugacy classes and centralisers of elements in such wreath
products explicitly. For finite wreath products, our approach yields efficient
algorithms for finding conjugating elements, conjugacy classes, and
centralisers.
