there are 6670903752021072936960 sudokugrids from
5472730538 equivalence classes ("S-classes")
where grids are S-equivalent, if they can be transformed into
each other by
permuting symbols
permuting the 3 rows in some stack
permuting the 3 columns in some band
permuting the 3 stacks
permuting the 3 bands
transposing
these 6 groups of transformations do commute and give
9!*6^8*2 transformations in total.
When we omit transposition, we get xxx "T-classes",
xxx S-classes have only 1 T-class, the other S-classes
have 2 T-classes.
there are only 306693 classes ("G-classes"), when we also allow
permuting the 3 entries in any minicolumn but remove transposition.
These 306693 classes generate 11555445688 sudokugrids,
by permuting minicolumns modulo permuting rows in a band,
at least one from each T-class. They can be generated in about
10? minutes but storing them requires too much memory.
Many sudoku-properties are an invariant of the S-class and thus
an invariant of the T-class and can be checked by checking
these 11555445688 generated sudokus.
(planning to fill in the missing numbers later...)
answers and updates to:
http://www.setbb.com/phpbb/viewtopic.php?t=194&mforum=sudoku