The GAP library implementation of the Dixon-Schneider algorithm first computes the linear characters, using the commutator factor group. If irreducible characters are missing afterwards, they are computed using the techniques described in Dix67, Sch90 and Hulpke93.
Called with a group G, the function CharacterTable
(see CharacterTable) returns a character table object that stores
already information such as class lengths, but not the irreducible
characters.
The routines that compute the irreducibles may use the information that
is already contained in this table object.
In particular the ordering of classes in the computed characters
coincides with the ordering of classes in the character table of G
(see The Interface between Character Tables and Groups).
Thus it is possible to combine computations using the group
with character theoretic computations
(see Advanced Methods for Dixon-Schneider Calculations for details),
for example one can enter known characters.
Note that the user is responsible for the correctness of the characters.
(There is little use in providing the trivial character to the routine.)
The computation of irreducible characters from the group needs to
identify the classes of group elements very often,
so it can be helpful to store a class list of all group elements.
Since this is obviously limited by the group order,
it is controlled by the global function IsDxLargeGroup
(see IsDxLargeGroup).
The routines compute in a prime field of size p, such that the exponent of the group divides (p-1) and such that 2 [Ö(|G|)] < p. Currently prime fields of size smaller than 65 536 are handled more efficiently than larger prime fields, so the runtime of the character calculation depends on how large the chosen prime is.
The routine stores a Dixon record (see DixonRecord) in the group
that helps routines that identify classes,
for example FusionConjugacyClasses, to work much faster.
Note that interrupting Dixon-Schneider calculations will prevent GAP
from cleaning up the Dixon record;
when the computation by IrrDixonSchneider is complete,
the possibly large record is shrunk to an acceptable size.
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GAP 4 manual