eleven wrote:Nevertheless it might be possible, that 3 symmetries already lead to the 648 automorphisms (?) Cant verify that now.

At least i can say, that the 3 symmetries in this grid

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`Most Canonical grid (normalised form)`

123456789

456789123

789123456

231564897

564897231

897231564

312645978

645978312

978312645

MR (123)(456)(789)

MD (159)(267)(348)

JR (147)(258)(369)

(and those you get by a common renumbering) only can be in equivalent grids (morphs of MC) and thus always imply all the other symmetries of MC.

This follows from

- 1 number in a band fixes the whole band (MR+MD the box, JR the other boxes)

- switching 2 bands preserves the 3 symmetries

- cycling the rows in a band preserves the 3 symmetries

- cycling the columns in all stacks the same way preserves the 3 symmetries

Then you always can transform a grid with these symmetries so, that you bring the 2 in the column of the 1 in box 1 (whereever it is) to the same minirow and -column in box 4 and same for the 3 in box 7 (23 cant be in the same minicolumn). Then cycle all rows in the bands and all columns in the stacks to move 1 to r1c1 and you must have the MC grid above.

But is it possible, that other grids have the 3 symmetries with other number cycles ? The numbers of one cycle in MR always have to be in different cycles of MD. Same must hold for MR/JR and MD/JR. This should mean, that (with given number cycles for MR/MD) only cycles for JR are possible, which correspond to changing the columns in 2 stacks - and maybe switching the stacks (giving equivalents to MC again).

So if i made no mistake, it is shown that the symmetries MR+MD+JR imply the symmetries

MC, JC, GR, GC, JD, HT, QT, DM, DDS, DM+JD, DM+MD, CS, RS, CS+MC, RS+MR, CS+JR, RS+JC, CS+GR, RS+GC

[Edit:]

This is not correct, see next page.