  We further develop the group-theoretic approach to fast matrix multiplication
introduced by Cohn and Umans, and for the first time use it to derive
algorithms asymptotically faster than the standard algorithm. We describe
several families of wreath product groups that achieve matrix multiplication
exponent less than 3, the asymptotically fastest of which achieves exponent
2.41. We present two conjectures regarding specific improvements, one
combinatorial and the other algebraic. Either one would imply that the exponent
of matrix multiplication is 2.
