  We present a new fast search algorithm for <m,m,m> Triple Product Property
(TPP) triples as defined by Cohn and Umans in 2003. The new algorithm achieves
a speed-up factor of 40 up to 194 in comparison to the best known search
algorithm. With a parallelized version of the new algorithm we are able to
search for TPP triples in groups up to order 55.
  As an application we identify a list of groups that would realize 5x5 matrix
multiplication with under 100 resp. 125 scalar multiplications (the best known
upper bound by Makarov 1987 resp. the trivial upper bound) if they contain a
<5,5,5> TPP triple. With our new algorithm we show that no group can realize
5x5 matrix multiplication better than Makarov's algorithm.
