This chapter describes the various group product constructions that are possible in GAP.
At the moment for some of the products methods are available only if both factors are given in the same representation or only for certain types of groups such as permutation groups and pc groups when the product can be naturally represented as a group of the same kind.
GAP does not guarantee that a product of two groups will be in a
particular representation. (Exceptions are WreathProductImprimitiveAction
and WreathProductProductAction which are construction that makes sense
only for permutation groups, see WreathProduct).
GAP however will try to choose an efficient representation, so products of permutation groups or pc groups often will be represented as a group of the same kind again.
Threfore the only guaranteed way to relate a product to its factors is via the embedding and projection homomorphisms (see Embeddings and Projections for Group Products);
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GAP 4 manual