Non-existence of a short algorithm for multiplication of $3\times 3$ matrices with group $S_4\times S_3$, II

It is proved that there is no an algorithm for multiplication of $3\times 3$ matrices of multiplicative length $\le 23$ 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 $(M_3(\mathbb{C}){)}^{\otimes 3}$ which was obtained earlier.