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What are the known bounds for the diameter of this subgroup? This thread gives a lower bound of 26 QTM but doesn't discuss HTM. I seem to recall a lower bound of 20 HTM from somewhere, but no upper bound. However, I recently found a position that requires 21 HTM:

U R F2 R U2 F2 U' F' R U F U R' F R2 U F R U2 R F'

Is this the first one known? Out of all 186624 positions with all pieces correctly permuted, this position and the inverse are the only ones requiring 21. Here's the full distribution:

Depth   Positions