Congratulations on your realization that there are EXACTLY 840 of what you call "configurations." This is exactly right, and it is what I meant by the following sentence in my Diamond Theory site --
"The 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2)."
A careful reader of my site would have found this in my remarks on the 4x4 case of the diamond theorem. I have no use for a file illustrating the 840 configurations, since they are easily obtained by substituting the 4 design elements in any of 4! = 24 ways in each of the 35 structures of the configurations that are illustrated in my Research Note 3, "Orthogonality of Latin Squares Viewed as Skewness of Lines." This is why I wrote "840 = 35 x 24."
It is clear from your statement that "the 840 configurations are a group" that you do not know what a group, in the technical mathematical sense, is.
I suggest you find out. Don't they teach mathematics at Brigham Young University?
