Geodesic
Non-existence of a short algorithm for multiplication of $3\times 3$ matrices whose group is $S_4\times S_3$,~II
Trudy Instituta matematiki, Tome 31 (2023) no. 1, pp. 101-111.

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It is proved that there is no algorithm for multiplication of $3\times 3$ matrices of multiplicative length $\leqslant 23$ that is invariant under a certain group isomorphic to $S_4\times S_3$. The proof uses description of the orbits of this group on decomposable tensors in the tensor cube $(M_3(\mathbb{C}))^{\otimes 3}$ which was obtained earlier. 
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V. P. Burichenko. Non-existence of a short algorithm for multiplication of $3\times 3$ matrices whose group is $S_4\times S_3$,~II. Trudy Instituta matematiki, Tome 31 (2023) no. 1, pp. 101-111. http://geodesic.mathdoc.fr/item/TIMB_2023_31_1_a11/

[1] V. P. Burichenko, “Non-existence of a short algorithm for multiplication of $3\times 3$ matrices with group $S_x\times S_3$”, Tr. In-ta matematiki [Proceedings of the Institute of mathematics], 30:1-2 (2022), 99–116

[2] V. P. Burichenko, Symmetries of matrix multiplication algorithms. I, 2015, arXiv: 1508.01110 | Zbl

[3] V. P. Burichenko, “The isotropy group of the matrix multiplication tensor”, Tr. In-ta matematiki [Proceedings of the Institute of mathematics], 24:2 (2016), 106–118 | MR

[4] R. P. Brent, Algorithms for matrix multiplication, Technical report 70–157, Stanford university, Computer Science Department, 1970 http://maths-people.anu.edu.au/brent/pub/pub002.html