These operations define a relation between equivalent grids. With respect to the 81 grid cell values, the rearranging operations form a subgroup of the symmetric group S81, of order 3!8×2 = 3,359,232. The relabeling operations are isomorphic with S9 and generate an additional 9! = 362,880 equivalent grids. Applying these operations on a grid results in 3!8×2×9! or 1,218,998,108,160 essentially equivalent grids. However, there is a small number of sudokus for which the above operations generate fewer grids; these are the self-similar, or automorphic sudokus. Only about 0.01% of all essentially unique grids are automorphic, but counting them is necessary for evaluating the exact number of essentially different sudokus.