🚀 go-pugleaf

J. McKinney asked for a reference to a list of all groups up to a given
limit. This question has several answers:

Listing *all* groups of a given order can be very tedious since there might
be lots of abelian groups of a given order, that can be obtained easily via
the canonical decomposition in elementary divisors. Thus most references
leave out abelian groups.

A standard reference (that should be aviable easily) is
   Coxeter, Moser: Generators and relations for discrete groups, 
   Ergebnisse der Mathematik und ihrer Grenzgebiete Bd. 14, Springer
that contains for example a list of all non abelian groups up to order 32.

If you would like to set the upper bound higher I strongly recommend
obtaining the group theoretical program GAP (aviable via anonymous ftp from

  samson.math.rwth-aachen.de
  dimacs.rutgers.edu
  math.ucla.edu
  pell.anu.edu.au

and various other servers), that contains (among others) all groups up to
order 100 and all 2-groups of order dividing 256.

Included are also character tables of a wide range of groups, including all
tables from the Cambridge ATLAS. The manual contains a more complete
bibliography of group lists.

The soon to be released (hopefully) version 3.2 will also include facilities
for computing the character table of a given finite group, which would allow
for a complete list of character tables of the groups up to order 100.

--
Alexander Hulpke,   Alexander.Hulpke@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany

In his article of 14-Sep-92 Jack McKinney writes:

         I am looking for a reference that contains a list of all the groups
    of all orders up to a given limit [i.e., a book can only be so large.]

The computer system GAP contains a collection of all solvable  groups  of
size up to 100.  The groups are represented by so called power-commutator
presentations.  This is  the reason why  we  didn't include A(5) for  the
collection of  *all* groups of size up to 100; A(5) cannot be represented
this way.  GAP is able to compute, among many other things, the character
tables of these groups in a moderate amount of time.

More information  about  GAP can be obtained by  getting 'pub/gap/README'
via anonymous 'ftp' to 'samson.math.rwth-aachen.de' (137.226.152.6).

The collection is a compilation of the data in

    Marshall Hall, Jr. and James K. Senior
    The Groups of Order 2^n (n <= 6)
    The Macmillan Company, New York, 1964
        [the groups of size 64]

    Reinhard Laue
    Zur Konstruktion und Klassifikation endlicher aufl"osbarer Gruppen
    vol. 9 of Bayreuther Mathematische Schriften, Uni. Bayreuth, 1982
        [the groups of size 96]

    Joachim Neub"user
    Die Untergruppenverb"ande der Gruppen der Ordnungen <= 100
     mit Ausnahme der Ordnungen 64 und 96
    Habilitationsschrift, Uni. Kiel, 1967
        [other sizes]

Martin.

--
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany

A.D. Thomas, G.V. Wood: Group Tables
Shiva Publishing Limited, 1980

The book contains all groups of order less than or equal to 32
and various information about them including the character
tables for the non-abelian groups.

Werner Nickel
Mathematics Research Section
Australian National University