Simple cycles in the $n$-cube with a large group of automorphisms

The cycle automorphism in the $n$-cube is the automorphism of the cube that leaves the cycle in place and does not change its orientation. An upper bound for the order of the group of cycle automorphisms in the $n$-cube is found. We obtain the construction for building long simple cycles for which the order of the group reaches the upper bound. Bibliogr. 3.