The finite irreducible linear 2-groups of degree 4

We present a classification of the finite irreducible 2-subgroups of GL(4, ℂ); that is, we give a parametrised list of representatives for the conjugacy classes of such groups. Each group listed is defined by a generating set of monomial matrices. There are essentially three possibilities for the projection of an irreducible monomial 2-group into the group of all permutation matrices. The classification problem accordingly falls into three separate cases. Each case may be handled by a general method consisting of three major steps. Techniques for applying the method to the most difficult case are developed in detail, so that the other two cases may then be dealt with routinely. The techniques used include elementary character theory, a method for drawing the Hasse diagram of the submodule lattice of a direct sum, and cohomology theory, particularly the calculation of 2-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence. Related questions concerning isomorphism between the listed groups, and Schur indices over ℚ, are also considered.