Decomposition Matrices

This is a database of decomposition matrices of "bicyclic" extensions of those finite simple groups for which the Atlas of Finite Groups contains the ordinary character tables. The data files are in PDF format.

Contents of this Database

For each simple group G of order less than 109 in the Atlas of Finite Groups, the Atlas of Brauer Characters shows the Brauer character tables for all primes dividing the order of G.

(The Atlas of Finite Groups was published by Oxford University Press. Currently the book is sold out, perhaps the publishers will produce a second edition if sufficiently many orders are sent to them.)

Also, for larger simple groups in the Atlas of Finite Groups, Brauer character tables (or equivalently, decomposition matrices) at least for some "bicyclic" extensions and some prime divisors of the group order are known. More precisely, the Brauer character tables for the groups listed on pages 102-155 of the Atlas of Finite Groups (including Co2) are known, except the few cases mentioned in the list of ongoing work. Note that so-called "generality problems" may occur for these groups.

For a list of people involved in the computations of Brauer character tables and decomposition matrices, see Section 17 in the Introduction to the Atlas of Brauer Characters. A list of references can be found here.

Decomposition Matrices in GAP

All tables contained in the Atlas of Brauer Characters and many others are available in the computer algebra system GAP, which provides many functions for computations with characters and character tables. This database has been created automatically from the character table library of GAP.

For dealing with decomposition matrices, GAP may be more suitable than the collection of files shown here in a quite restricted way. The decomposition matrices can be easily derived from the ordinary and Brauer character table, a sample GAP session is available in HTML and PDF format.

Status of this Database

If you have suggestions how this database can be improved then feel free to send an e-mail message to .

This may concern for example known decomposition matrices that are missing here, possible errors in character tables and decomposition matrices, the layout of the pages, or additional related information (such as Brauer trees) that may be added to the database.

Structure of this Database

Suppose you are interested in the p-modular decomposition matrices of a bicyclic extension m.G.a of the group G. Then you first choose the group G in the table shown below, according to the name used for G in the Atlas of Finite Groups. From the documents available for G, you then choose the one you need according to its filename, which is composed from the name for G.a and the suffix modp.pdf.

The decomposition matrices are shown blockwise. For symmetric groups, alternatively, the full p-modular decomposition matrix is also available as a lower triangular matrix whose rows are labelled by partitions. The files containing this format have suffixes partmodp.pdf.

Here are a few examples.
The 2-modular decomposition matrices of the Mathieu group M24 are in the directory M24, in the file M24mod2.pdf.
The PDF file containing the 5-modular decomposition matrices of the McLaughlin group McL and its triple cover 3.McL is McLmod5.pdf in the directory McL.
The PDF file containing the 3-modular decomposition matrices of the symmetric group S5 and its double cover 2.S5 is A5.2mod3.pdf in the directory A5. (Note that the decomposition matrices are independent of the choice of the double cover.)
The PDF file containing the full 2-modular decomposition matrix of the symmetric group S7, with rows labelled by partitions, is A7.2partmod2.pdf in the directory A7.

A description of the format of the document files is available as a PDF file.

The Groups

2E6(2) 2F4(2)' 3D4(2) A5 A6 A7 A8 A9 A10 A11
A12 A13 A14 A15 A16 A17 A18 A19 B Co1
Co2 Co3 F3+ F4(2) Fi22 Fi23 G2(3) G2(4) G2(5) HN
HS He J1 J2 J3 J4 L2(8) L2(11) L2(13) L2(16)
L2(17) L2(19) L2(23) L2(25) L2(27) L2(29) L2(31) L2(32) L2(49) L2(81)
L3(2) L3(3) L3(4) L3(5) L3(7) L3(8) L3(9) L4(3) L4(4) L5(2)
L6(2) Ly M M11 M12 M22 M23 M24 McL O7(3)
O8+(2) O8+(3) O8(2) O8(3) O10+(2) O10(2) ON R(27) Ru S4(4)
S4(5) S6(2) S6(3) S8(2) S10(2) Suz Sz(8) Sz(32) Th U3(3)
U3(4) U3(5) U3(7) U3(8) U3(9) U3(11) U4(2) U4(3) U5(2) U6(2)

File created on 27-May-1999 by Thomas Breuer.
Last modified on 09-November-2020 by Thomas Breuer.