For tower groups (finite direct products of S_k^{a_k} with k >= 3), LatAut of the product is S_{a_4} times a symmetric group on the remaining factors, and the LatAut tower terminates at the trivial group after exactly three steps at most.
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Termination of the Lattice-Automorphism Tower for Direct Products of Symmetric Groups
For tower groups (finite direct products of S_k^{a_k} with k >= 3), LatAut of the product is S_{a_4} times a symmetric group on the remaining factors, and the LatAut tower terminates at the trivial group after exactly three steps at most.