61.7 DCE Presentations

The input to the GAP Double Coset Enumerator is presented as a record. This has the following compulsory components.

groupK:

The group K, given as a GAP group. In general, it is best to represent K as a permutation group of low degree.

gainGroups:

This specifies the types (K-conjugacy classes) of gain groups L associated with the generators. It takes the form of a list of records, each with the following components:

dom -- A representative of a set on which K acts in the same way that it acts on the left cosets of L. If this is not given then L=K and other fields are set accordingly.

op -- The operation of K on this set. This should be a GAP operation such as OnPoints. If op is not given, and dom is an integer then op defaults to OnPoints. If op is not given and dom is a set, then the op defaults to OnSets.

gens:

This field specifies the generators (the set X). It is a list of records, each with the fields:

name -- The abstract generator that will be used to denote this generator in the relations and subgroup generators.

invol -- A Boolean value indicating whether this generator should be considered as its own inverse. Default false.

inverse -- The 'name' of the inverse of this generator. This field is ignored if invol is present. If both inverse and invol are absent then a new generator will be created to be an inverse.

wgg -- The index (in gainGroups) of the gain group of this generator (up to conjugacy).

ggconj -- The gain group conjugator. The actual gain group of this generator will be that defined by entry wgg of the gainGroups list, conjugated by the element ggconj (of K). If this field is absent then it is taken to be the identity of K.

action -- This specifies the isomorphism theta_x induced by x between L_x and L_{x^{-1}}. It can be false, indicating no action, an element of K, indicating action by conjugation, or it can be an explicit isomorphism. The default is false. If an explicit homomorphism is given and the the field invol is not present, then the field inverse must be present; that is, a generator inverse to x cannot be synthesized in this case.

relators:

The relations of the presentation, as a list of DCE Words. Certain additional fields may be added to the words (which are represented as records) to optimize the calculation. These are described below.

subgens:

The generators of H, as a list of DCE Words.

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