  
  [1X5 [33X[0;0YOperations[133X[101X
  
  [33X[0;0YThe  generic representation of wreath product elements in wreath products of
  finite  groups  and in particular their (sparse) wreath cycle decompositions
  can be used to speed up certain computations in wreath products.[133X
  
  [33X[0;0YIn   particular   this   package  provides  efficient  methods  for  finding
  conjugating    elements,    conjugacy   classes,   and   centralisers.   The
  implementations are based on [BNRW22] and references therein.[133X
  
  
  [1X5.1 [33X[0;0YOperations List[133X[101X
  
  [33X[0;0YHere  we  include  a  list  of operations that take advantage of the generic
  representation of wreath product elements.[133X
  
  [33X[0;0YWe  include  python scripts in the [10Xdev/[110X directory that benchmark the [5XWPE[105X and
  native [5XGAP[105X implementations of these operations separately. The comparison of
  the  runtimes  supports  the  conclusion that the [5XWPE[105X implementations are an
  order  of  magnitude  faster than the native [5XGAP[105X implementations. We can now
  solve  these  computational  tasks  for  large  wreath  products  that  were
  previously not feasible in [5XGAP[105X[133X
  
  
  [1X5.1-1 [33X[0;0YWreath Product Representations[133X[101X
  
  [33X[0;0YIn the following let [22XG = K ≀ H[122X be a wreath product, where [22XH ≤ Sym(m)[122X.[133X
  
  [33X[0;0YIn  [5XGAP[105X  the  wreath  product  [22XG[122X  can  be  given  in  one  of  the following
  representations :[133X
  
  [30X    [33X[0;6YGeneric Representation[133X
  
  [30X    [33X[0;6YPermutation Representation in Imprimitive Action[133X
  
  [30X    [33X[0;6YPermutation Representation in Product Action[133X
  
  [30X    [33X[0;6YMatrix Representation[133X
  
  
  [1X5.1-2 [33X[0;0YOperations for all Representations[133X[101X
  
  [33X[0;0YFurther let [22Xx, y ∈ P = K ≀ Sym(m)[122X be elements of the parent wreath product [22XP[122X
  which is given in the same representation as [22XG[122X.[133X
  
  [33X[0;0YThe  following  operations  use  implementations  that  exploit  the generic
  representation and (sparse) wreath cycle decompositions :[133X
  
  [30X    [33X[0;6Y[10XRepresentativeAction(G, x, y)[110X[133X
  
  [30X    [33X[0;6Y[10XCentraliser(G, x)[110X[133X
  
  [30X    [33X[0;6Y[10XConjugacyClasses(G)[110X[133X
  
  
  [1X5.1-3 [33X[0;0YOperations for Permutation Representations[133X[101X
  
  [33X[0;0YHere we assume that [22XG[122X is given in some permutation representation.[133X
  
  [33X[0;0YThe  following  operations  use  implementations  that  exploit  the generic
  representation and (sparse) wreath cycle decompositions :[133X
  
  [30X    [33X[0;6Y[10XCycleIndex(G)[110X[133X
  
  
  [1X5.1-4 [33X[0;0YOperations for Generic Representation[133X[101X
  
  [33X[0;0YHere we assume that [22XG[122X is given in generic representation.[133X
  
  [33X[0;0YThe  following  operations  use  implementations  that  exploit  the generic
  representation and (sparse) wreath cycle decompositions :[133X
  
  [30X    [33X[0;6Y[10XOrder(x)[110X[133X
  
  [33X[0;0Y [133X
  
