The GAP Library of Tables of Marks

Version 1.2.11

Released 2024-01-12

This project is maintained by Liam Naughton, Götz Pfeiffer

This is the GAP Library of Tables of Marks, which is an add-on package for GAP 4.

From the Table of marks library documentation:

The concept of a

table of markswas introduced by W. Burnside in his 1955 bookTheory of Groups of Finite Order. Therefore a table of marks is sometimes called a Burnside matrix. The table of marks of a finite groupGis a matrix whose rows and columns are labelled by the conjugacy classes of subgroups ofGand where for two subgroupsHandKthe(H, K)–entry is the number of fixed points ofKin the transitive action ofGon the cosets ofHinG. So the table of marks characterizes the set of all permutation representations ofG. Moreover, the table of marks gives a compact description of the subgroup lattice ofG, since from the numbers of fixed points the numbers of conjugates of a subgroupKcontained in a subgroupHcan be derived. For small groups the table of marks ofGcan be constructed directly in GAP by first computing the entire subgroup lattice ofG. However, for larger groups this method is unfeasible. The GAP Table of Marks library provides access to several hundred tables of marks and their maximal subgroups.

If you have found important features missing or if there is a bug, let us know and we will try to address it in the next version of the GAP Library of Tables of Marks. Please either use the our issue tracker or send a short email to Liam Naughton l.naughton@wlv.ac.uk.

This holds in particular if you have used the GAP Library of Tables of Marks to solve a problem.

You can also reach the GAP developers by sending an email to the general GAP email address support@gap-system.org for problems with GAP.

You may have got the GAP Library of Tables of Marks as or a compressed tar archive (filename ends with .tar.gz). Use the appropriate command on your system to unpack the archive.

On UNIX systems the compressed tar archive may be unpacked by

```
tar xzf tomlib-x.y.z.tar.gz
```

or, if tar on your system does not understand the option z, by

```
gunzip tomlib-x.y.z.tar.gz
tar xf tomlib-x.y.z.tar
```

Installation of the GAP Library of Tables of Marks means unpacking the archive file in an appropriate directory.

For the two ways of installing GAP 4 packages in general, see the sections “Installing a GAP Package” and “GAP Root Directory” in the GAP 4 Reference Manual.

Once you have successfully installed the GAP Library of Tables of Marks, it is usually loaded automatically when GAP is started; if not then it can be loaded explicitly by typing

```
gap> LoadPackage( "tomlib" );
true
```

at the GAP prompt. Then the functions of the library are available in the current session.