  It is proved that there is no an algorithm for multiplication of $3\times3$
matrices of multiplicative length $\leq23$ that is invariant under a certain
group isomorphic to $S_4\times S_3$. The proof makes use of description of the
orbits of this group on decomposable tensors in the tensor cube $(M_3({\mathbb
C}))^{\otimes3}$ which was obtained earlier.
