Maximizing the mutual information of high rate linear matrix modulation schemes for MIMO channels is considered. Linear matrix modulators are described in terms of a space-time basis, which is a three-index object, one index for the symbol, one for time and one for transmit antennas. Corresponding to these, there are three kinds of transformations that leave mutual information invariant; orthogonal symbol rotations, unitary time and unitary antenna rotations. If a group structure is required of the parts of these transformations that embed into the other, the space-time basis is a set of Clebsch-Gordan coefficients realizing the equivalence of representations of this group, and the space-time basis is an extremum of second-order mutual information. Thus, maximal mutual information matrix modulators may be found using the well developed theory of group representations. As an example, symbol rate 1.5 schemes for four transmit antennas, extending over four channel uses, are considered.