Nesting of multiplicative tables

Multiplicative tables

Tables T(k) of numbers m*n mod k, e.g.:

T1      T2     T3     T4          T5
        1       1 2        1 2 3       1 2 3 4
                2 1        2 0 2       2 4 1 3
                           3 2 1       3 1 4 2
                                       4 3 2 1
T6                T7                  T8
  1 2 3 4 5        1 2 3 4 5 6         1 2 3 4 5 6 7
  2 4 0 2 4        2 4 6 1 3 5         2 4 6 0 2 4 6
  3 0 3 0 3        3 6 2 5 1 4         3 6 1 4 7 2 5
  4 2 0 4 2        4 1 5 2 6 3         4 0 4 0 4 0 4
  5 4 3 2 1        5 3 1 6 4 2         5 2 7 4 1 6 3
                   6 5 4 3 2 1         6 4 2 0 6 4 2
                                       7 6 5 4 3 2 1
T9                        T10
 1  2 3 4  5 6 7  8           1 2 3  4 5 6  7 8 9
 2  4 6 8  1 3 5  7           2 4 6  8 0 2  4 6 8
 3  6 0 3  6 0 3  6           3 6 9  2 5 8  1 4 7
 4  8 3 7  2 6 1  5           4 8 2  6 0 4  8 2 6
 5  1 6 2  7 3 8  4           5 0 5  0 5 0  5 0 5
 6  3 0 6  3 0 6  3           6 2 8  4 0 6  2 8 4
 7  5 3 1  8 6 4  2           7 4 1  8 5 2  9 6 3
 8  7 6 5  4 3 2  1           8 6 4  2 0 8  6 4 2
                              9 8 7  6 5 4  3 2 1
T11                               T12
  1  2  3  4  5  6  7  8  9 10     1  2  3  4  5  6  7  8  9 10 11
  2  4  6  8 10  1  3  5  7  9     2  4  6  8 10  0  2  4  6  8 10
  3  6  9  1  4  7 10  2  5  8     3  6  9  0  3  6  9  0  3  6  9
  4  8  1  5  9  2  6 10  3  7     4  8  0  4  8  0  4  8  0  4  8
  5 10  4  9  3  8  2  7  1  6     5 10  3  8  1  6 11  4  9  2  7
  6  1  7  2  8  3  9  4 10  5     6  0  6  0  6  0  6  0  6  0  6
  7  3 10  6  2  9  5  1  8  4     7  2  9  4 11  6  1  8  3 10  5
  8  5  2 10  7  4  1  9  6  3     8  4  0  8  4  0  8  4  0  8  4
  9  7  5  3  1 10  8  6  4  2     9  6  3  0  9  6  3  0  9  6  3
 10  9  8  7  6  5  4  3  2  1    10  8  6  4  2  0 10  8  6  4  2
                                  11 10  9  8  7  6  5  4  3  2  1

Self instances

Here we can select also (similarly as in G-systems) such numbers, which are self, not nested.
(See above - rows and columns, which do not have number 0.) Tables of self instances have φ(k) rows and columns, where φ is Euler function.

T1      T2      T3       T4        T5
        1        1 2        1 3          1 2 3 4
                 2 1        3 1          2 4 1 3
                                         3 1 4 2
                                         4 3 2 1
T6     T7               T8
 1 5        1 2 3 4 5 6         1 3 5 7
 5 1        2 4 6 1 3 5         3 1 7 5
            3 6 2 5 1 4         5 7 1 3
            4 1 5 2 6 3         7 5 3 1
            5 3 1 6 4 2
            6 5 4 3 2 1
T9              T10
 1 2 4 5 7 8        1 3 7 9
 2 4 8 1 5 7        3 9 1 7
 4 8 7 2 1 5        7 1 9 3
 5 1 2 7 8 4        9 7 3 1
 7 5 1 8 4 2
 8 7 5 4 2 1
T11                              T12
  1  2  3  4  5  6  7  8  9 10      1  5  7 11
  2  4  6  8 10  1  3  5  7  9      5  1 11  7
  3  6  9  1  4  7 10  2  5  8      7 11  1  5
  4  8  1  5  9  2  6 10  3  7     11  7  5  1
  5 10  4  9  3  8  2  7  1  6
  6  1  7  2  8  3  9  4 10  5
  7  3 10  6  2  9  5  1  8  4
  8  5  2 10  7  4  1  9  6  3
  9  7  5  3  1 10  8  6  4  2
 10  9  8  7  6  5  4  3  2  1

Schematic algebra