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Contents of Database of Groups of Order Dividing 729


Contents of Database of Groups of Order Dividing 729




This database contains files storing compact descriptions 
for the groups of order dividing 729.  The files are:

thrgp3          thrgp27         thrgp243
thrgp9          thrgp81         thrgp729




A decoding procedure, gen3, is also supplied with the database.


There are 594 groups of order 729 precisely. The number of 
d-generator groups for each d in {1, ..., 6} is given below:

d = 1        1 group
d = 2        2 groups
d = 3        5 groups
d = 4       15 groups
d = 5       67 groups
d = 6      504 groups




The groups are arranged as a sequence gps. The following criteria are
used, in turn, to determine the index of a group in the sequence:

    

  1. 
increasing generator number (the generator number d of a group G
is the smallest cardinality of a set of generators of G);

  2. 
increasing exponent-p class;

  3. 
the position of its parent in the sequence of the appropriate order;

  4. 
the sequence in which the implementation of the p-group generation
algorithm outputs the immediate descendants of a group.


Further information on the organisation of the groups and explanations
of the technical terms used here can be found in the research papers
(see Bibliography for Database of Groups of Order Dividing 729). Knowledge of this 
organisation can be used to determine the selection 
and processing of groups.


The compact description for a group encodes the exponents of the P-CP
of the group. There are (n-1)*n*(n+1)/6 such exponents for a group of
order 3^n, each of which is 0, 1, or 2. An examination of the text
of the procedure gen3 will reveal the encoding method used. Each
compact description is a sequence of 5 integers. For example the stored
description of the 57th group of order 729 is:

[ 2, 6, 3, 4, 24669014107005 ]


The first entry 2 is the generator number; the second 6 is the number
of P-CP generators for the group; the third 3 is the prime; the fourth
is the exponent-3 class of the group; the last integer encodes the 
35 exponents for the P-CP. The compact description of each group is 
stored as an element of a sequence gps.



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