Finite symplectic matrix groups

A database of all symplectic irreducible maximal finite subgroups of GL(2n, Q) up to n=11 is included in Magma (starting from version 2.16.4). Note that it must be downloaded seperately from the Magma download page, just like all optional databases.

Alternatively, the file sgroups.m contains invariant forms for all symplectic primitive irreducible maximal finite subgroups of GL(2n, Q) up to n=11.

It is designed to be used with the computer algebra system Magma (version 2.14 or higher). Since it contains only ascii text, it can be opened in any text editor. So it should be straightforward to adopt the file for the computer algebra system of your choice.

How to use this file with Magma:

Load the file in magma via


  load "sgroups.m";

This generates an associative array called SGroups.

The data for dimension 2n is stored in SGroups[2n] as a sequence of triples <FF, o, s>. The sequence is ordered in the same way as in the thesis.

FF is a sequence of two forms F,S such that A:= AutomorphismGroup(FF); generates a representative for the corresponding conjugacy class.
The forms F,S are normalized as described in the appendix.
o is the order of the groups in the conjugacy class
s is a string containing the name of this class given in my thesis (\tensor is abbreviated t and so on...)

Example:

Suppose we want to access the groups GL(2,3) < GL(4, Q). Either from the thesis or from


  [x[3] : x in SGroups[4]];
  
  which yields the list:
  
  > [ D8tC4.S3, C4tA2, GL23, SL23oC3, C10 ]

we know that it is the third entry of SGroups[4]. So


  A:= AutomorphismGroup(SGroups[4,3,1]);

generates a representative A of that class.