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

Non-existence of a short algorithm for multiplication of $3\times3$ matrices with group $S_4\times S_3$, II