The structure of the projective indecomposable modules of the Mathieu group \(M_{12}\) in characteristic 2 and 3 (Q689695)

In this note the author determines the socle structure of the projective indecomposable modules for the Mathieu group \(M_{12}\) in characteristic 2 and 3. So as byproduct \(\text{Ext}^ 1\) for simple modules can be read off directly. The proofs are based on computer computations using mainly CAYLEY and MEATAXE. For the determination of the socle structure algorithms are used which have been developed by the author himself [J. Symb. Comput. 9, 607- 636 (1990; Zbl 0703.20009); Computing with homomorphisms of modular representations (preprint)]. In the meantime these are also part of CAYLEY.