  We describe certain special consequences of certain elementary methods from
group theory for studying the algebraic complexity of matrix multiplication, as
developed by H. Cohn, C. Umans et. al. in 2003 and 2005. The measure of
complexity here is the exponent of matrix multiplication, a real parameter
between 2 and 3, which has been conjectured to be 2. More specifically, a
finite group may simultaneously "realize" several independent matrix
multiplications via its regular algebra if it has a family of triples of
"index" subsets which satisfy the so-called simultaneous triple product
property (STPP), in which case the complexity of these several multiplications
does not exceed the rank (complexity) of the algebra. This leads to bounds for
the exponent in terms of the size of the group and the sizes of its STPP
triples, as well as the dimensions of its distinct irreducible representations.
Wreath products of Abelian with symmetric groups appear especially important,
in this regard, and we give an example of such a group which shows that the
exponent is less than 2.84, and could be possibly be as small as 2.02 depending
on the number of simultaneous matrix multiplications it realizes.
