On automorphism group of a possible short algorithm for multiplication of $3\times 3$ matrices

Studying algorithms admitting nontrivial symmetries is a prospective way of constructing new short algorithms of matrix multiplication. The main result of the article is that if there exists an algorithm of multiplicative length $l\le 22$ for multuplication of $3\times 3$ matrices then its automorphism group is isomorphic to a subgroup of $S_l\times S_3$.