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This paper describes the socle series of the projective indecomposable modules and of tensor products of simple modules for the simple group Sz(8) in characteristic 2. The results have been obtained by computational means and the various steps are described. The main algorithm was modified to allow for parallel execution on a network of workstations. This made possible the effective handling of modules of degree 4030. In this note we want to describe how to determine the Loewy series of the projective indecomposable modules of the smallest Suzuki group of order 29120 = 26 • 5 • 7 • 13 in characteristic 2. The computer will be the main tool used to achieve this goal, in particular, the Cayley system (see [1]) will be applied with the author's implementation of various representation-theoretic algorithms. These algorithms have been explained in detail in [6, 7]. This shows how the computer helps to gain insight into examples that seem to be difficult to analyze by purely theoretical means, once the required methods are available and implemented. The notation used in this paper is standard (see [6, 4] for further information). For convenience, we include both the ordinary and 2-modular character table of this group in the appendix (see the Supplement section at the end of this issue). The tables are taken from [3, 5]. The group Sz(8) has eight modular characters. Seven lie in the principal block and are denoted by /, 4a , 4i,, 4C, 16a , 16ft, and 16C. The module 64 lies in a block of defect 0. With the help of the information in the character tables the computation of the decomposition and Cartan matrix is a standard task, and the results are included in the appendix as well. Since the computer is used to do most of the work, the representations have to be constructed explicitly. The four-dimensional representations are easy to obtain, as they are described in the Atlas. The field GF(8) is the smallest field over which all representations can be realized. The 64-dimensional representation can be constructed from the permutation representation of Sz(8) on 65 letters (which is provided as part of the Cayley library facility), and thus it can be realized over GF(2). While the naming of the four-dimensional modules may be arbitrary, some care has to be used in distinguishing the 16-dimensional Received by the editor November 12, 1991. 1991 Mathematics Subject Classification. Primary 20C20, 20C40, 20C05, 20H20. ©1993 American Mathematical Society 0025-5718/93 $1.00+ $.25 per page

The structure of the projective indecomposable modules of the Suzuki group 𝑆𝑧(8) in characteristic 2