I believe the 6x6 case has no solutions. The proof is the 36 Officer Problem.
How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 6x6 array such that no row or column duplicates a rank or a regiment? The answer is that no such arrangement is possible.
Suppose they were trying to form lunch bunches: the first arrangement is all the same rank, the second is all the same regiments. Then we know there is no third arrangement and the schedule fails.
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