Comparison with the ATLAS table
The ATLAS of finite groups (Conway, et. al, 1985) is a reference work that
contains information about many simple groups and their close relatives.
Amongst other information, it gives the character table of the group. The page
for the same group M11 is reproduced here. The letter notation for cyclotomic
irrationalities origins in this book and is explained in the preface.
The “Construction”
section describes some
ways how to obtain
the group, from
scratch or within other
groups. A presentation
is a set of defining rules
for the group, i.e. any
relation among
elements (as products
in generators) is a
consequence.
For each (class of ) maximal
subgroups a representative is
listed, together with some
structure information, the
decomposition of the
permutation character on the
cosets, and ways to construct
such subgroups in the full
group.
The indicator
column gives
the
FrobeniusSchurIndicator (see
future
homework)
that describes
whether the
corresponding
representation
is real.
The layout of the table
(10 classes here). This
is mainly relevant if
the table for the group
and some extensions
are given together.
There is no general way to assign names to groups that determine isomorphism
type. However sometimes there are just few (or even only one) group with a
given composition structure, this happens to hold for many of the more
interesting groups. The ATLAS notation uses the form A.B to denote an
extension of a normal subgroup A by a factor group B. If the extension is known
to be split (semidirect product) a colon : is used. If the extension is known to be
not split a high dot ˙is used.
Classes are labeled by
element order and
letters. A name B**
indicates that this is a
class of suitable powers
of the corresponding Aclass.
The p-power maps give
the classes of powers for
the primes dividing the
order, i.e. AB for a class of
order 6 means that g2 is
in class 3A and g3 in class
3B.