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The GAP group collection contains all character tables that are included in the Atlas of finite groups (CCN85, from now on called ATLAS) and the Brauer tables contained in the modular ATLAS (JLPW95). Although the Brauer tables form a library of their own, they are described here since all conventions for ATLAS tables stated here hold for Brauer tables, too.
Additionally some conventions are necessary about follower characters!
These tables have the information
origin: ATLAS of finite groups
resp.
origin: modular ATLAS of finite groups
in their text component, further on they are simply called ATLAS
tables.
In addition to the information given in Chapters 6--8 of the ATLAS which tell how to read the printed tables, there are some rules relating these to the corresponding GAP tables.
Improvements
Note that for the GAP library not the printed ATLAS is relevant but the revised version given by the list of Improvements to the ATLAS which can be got from Cambridge.
Also some tables are regarded as ATLAS tables which are not printed in the ATLAS but available in ATLAS format from Cambridge; at the moment, these are the tables related to L_2(49), L_2(81), L_6(2), O_8^-(3), O_8^+(3) and S_{10}(2).
Powermaps
In a few cases (namely the tables of 3.McL, 3_2.U_4(3) and its
covers, 3_2.U_4(3).2_3 and its covers) the powermaps are not uniquely
determined by the given information but determined up to matrix
automorphisms (see MatAutomorphisms) of the characters; then the first
possible map according to lexicographical ordering was chosen, and the
automorphisms are listed in the text component of the concerned table.
Projective Characters
For any nontrivial multiplier of a simple group or of an automorphic
extension of a simple group, there is a component projectives in the
table of G that is a list of records with the names of the covering
group (e.g. "12_1.U4(3)") and the list of those faithful characters
which are printed in the ATLAS (so--called it proxy characters).
Projections
ATLAS tables contain the component projections: For any covering
group of G for which the character table is available in ATLAS
format a record is stored there containing components name (the name of
the cover table) and map (the projection map); the projection maps any
class of G to that preimage in the cover for that the column is printed
in the ATLAS; it is called g_0 in Chapter 7, Section 14 there.
(In a sense, a projection map is an inverse of the factor fusion from the cover table to the actual table (see ProjectionMap).)
Tables of Isoclinic Groups
As described in Chapter 6, Section 7 and Chapter 7, Section 18 of the
ATLAS, there exist two different groups of structure 2.G.2 for a
simple group G which are isoclinic. The ATLAS table in the library
is that which is printed in the ATLAS, the isoclinic variant can be got
using CharTableIsoclinic CharTableIsoclinic.
Succession of characters and classes
(Throughout this paragraph, G always means the involved simple group.)
item For G itself, the succession of classes and characters in the
GAP table is as printed in the ATLAS.
item For an automorphic extension G.a, there are three types of
characters:
item If a character chi of G extends to G.a, the different
extensions chi^0,chi^1,ldots,chi^{a-1} are consecutive (see
ATLAS, Chapter 7, Section 16).
item If some characters of G fuse to give a single character of G.a,
the position of that character is the position of the first
involved character of G.
item If both, extension and fusion, occur, the result characters are
consecutive, and each replaces the first involved character.
item Similarly, there are different types of classes for an automorphic
extension G.a:
item If some classes collapse, the result class replaces the first
involved class.
item For a > 2, any proxy class and its followers are consecutive; if
there are more than one followers for a proxy class (the only case
that occurs is for a = 5), the succession of followers is the
natural one of corresponding galois automorphisms (see ATLAS,
Chapter 7, Section 19).
The classes of G.a_1 always precede the outer classes of G.a_2
for a_1, a_2 dividing a and a_1 < a_2. This succession is
like in the ATLAS, with the only exception U_3(8).6.
item For a central extension M.G, there are different types of
characters:
item Every character can be regarded as a faithful character of the
factor group m.G, where m divides M. Characters faithful for
the same factor group are consecutive like in the ATLAS, the
succession of these sets of characters is given by the order of
precedence 1, 2, 4, 3, 6, 12 for the different values of m.
item If m > 2, a faithful character of m.G that is printed in the
ATLAS (a so-called mboxem proxy) represents one or more
mboxem followers, this means galois conjugates of the proxy;
in any GAP table, the proxy precedes its followers; the case
m = 12 is the only one that occurs with more than one follower
for a proxy, then the three followers are ordered according to the
corresponding galois automorphisms 5, 7, 11 (in that succession).
item For the classes of a central extension we have:
item The preimages of a G-class in M.G are subsequent, the
succession is the same as that of the lifting order rows in the
ATLAS.
item The primitive roots of unity chosen to represent the generating
central element (class 2) are E(3), E(4), E(6)^5 (= E(2)
* E(3)) and E(12)^7 (= E(3) * E(4)) for m = 3, 4, 6 and
12, respectively.
item For tables of bicyclic extensions m.G.a, both the rules for
automorphic and central extensions hold; additionally we have:
item Whenever classes of the subgroup m.G collapse or characters fuse,
the result class resp. character replaces the first involved class
resp. character.
item Extensions of a character are subsequent, and the extensions of a
proxy character precede the extensions of its followers.
item Preimages of a class are subsequent, and the preimages of a proxy
class precede the preimages of its followers.
GAP 3.4.4