7.103 WreathProduct

WreathProduct( G, H )
WreathProduct( G, H, alpha )

In the first form of WreathProduct the right regular permutation representation of H on its elements is used as the homomorphism alpha. In the second form alpha must be a homomorphism of H into a permutation group. Let d be the degree of the range of alpha. Then WreathProduct returns the wreath product of G by H with respect to alpha, that is the semi-direct product of the direct product of d copies of G which are permuted by H through application of alpha to H.

    gap> s3 := Group( (1,2,3), (1,2) );
    Group( (1,2,3), (1,2) )
    gap> z2 := CyclicGroup( AgWords, 2 );
    Group( c2 )
    gap> f := IdentityMapping( s3 );
    IdentityMapping( Group( (1,2,3), (1,2) ) )
    gap> w := WreathProduct( z2, s3, f );
    Group( WreathProductElement(
    c2, IdAgWord, IdAgWord, (), () ), WreathProductElement( IdAgWord,
    c2, IdAgWord, (), () ), WreathProductElement( IdAgWord, IdAgWord,
    c2, (), () ), WreathProductElement( IdAgWord, IdAgWord, IdAgWord,
    (1,2,3),
    (1,2,3) ), WreathProductElement( IdAgWord, IdAgWord, IdAgWord, (1,2),
    (1,2) ) )
    gap> Factors( Size( w ) );
    [ 2, 2, 2, 2, 3 ] 

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