It is possible to study the complexity of complex matrix multiplication maps (bilinear maps or tensors) via finite groups which realize certain special subset triples, called triple product property (TPP) triples.  The multiplicative size |S|*|T|*|U| of a TPP triple (S,T,U) realized by a finite group G determines "volume" or "size" of the corresponding tensor describing |S| x |T| by |T| x |U| matrix multiplication.  For a fixed group G the tensor of largest size realized by it determines its support for rectangular matrix multiplication, and in general, the focus is on finding families of finite groups which realize maximally sized tensors with respect to the group sizes.

TPP triples allow matrices to be injectively embedded into the regular group algebras of their host finite groups, and to be multiplied and their matrix products to be recovered fully from group algebra products via a group-theoretic matrix multiplication algorithm.  This allows us to describe the parameters of the complexity of matrix multiplication, such as tensor rank and the exponent of matrix multiplication, in terms of the sizes of the TPP triples, and of the sizes of their host groups and other group-theoretic and representation-theoretic parameters. The optimal groups for matrix multiplication, it appears, are those which, with respect to their sizes, realize large volumes of matrix multiplication but have small irreducible representation sizes.

TPP triples are structured and built up in very definite and interesting ways within finite groups, and a (1) number of new combinatorial and algebraic results about them are presented here, and the applications of such results in some dihedral groups, alternating groups, and wreath product groups. Also presented are (3) programs written and implemented on the GAP computer algebra system, which search for these TPP triples for arbitrary input finite groups, and an analysis of their complexity, and (4) a small database of results obtained for small non-Abelian groups of order upto 60.