Double Coset Enumeration (DCE) can be seen either as a space- (and time-) saving variant of ordinary Coset Enumeration (the Todd-Coxeter procedure), as a way of constructing finite quotients of HNN-extensions of known groups or as a way of constructing groups given by symmetric presentations in a sense defined by Robert Curtis. A double coset enumeration works with a finitely-presented group G, a finitely generated subgroup H (given by generators) and a finite subgroup K, given explicitly, usually as a permutation group. The action of G on the cosets of H divides into orbits under K, and is constructed as such, using only a relatively small amount of information for each orbit.
The next two sections Authorship and Contact Information and Installing the DCE Package describe the authorship of the package, and the simple procedure for installing it.
In Mathematical Introduction the calculation performed by the double coset enumerator, and the meaning of the input is described more DCE Words and DCE Presentations describe how the input is organized as Examples of Double Coset Enumeration.
The data structure returned by DCE is described in The DCE Universe and Informational Messages from DCE. Succeeding sections: DCE, DCESetup, DCEPerm, DCEPerms, DCEWrite and DCERead describe the basic functions used to run DCE, extract information from the result, and save and restore double Example of DCE Functions.
The user can exert considerable control over the behaviour of DCE, as Example of Double Coset Enumeration Strategies.
Since double coset enumeration can construct permutation representations of very high degree, it may not be feasible to extract permutations from the result. Nevertheless, some analysis of the permutation representation Functions for Analyzing Double Coset Tables and the functions used are documented in: DCEColAdj, Example of DCEColAdj.
Finally, the link with Robert Curtis' notion of a symmetric Double Coset Enumeration and Symmetric Presentations with detailed documentation in Examples of DCE and Symmetric Presentations.
More detailed documentation of the data structures used in double coset
enumeration, and the internal functions available to access them is found
in the document ``GAP Double Coset Enumerator -- Internals'', found in
the doc directory of the dce package.
GAP 3.4.4