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Complement Pair Pattern Groups

The groups are classified by the complement pair patterns of the squares.
The lines joining the complement pairs are not shown here.
Pair numbers are indicated by the same letter, (a, b, ...).

Group Form

The group form of a square is determined as follows:

• the square is listed as a 1-dimensional array of 25 cells, indexed 0 to 24
• a sequence is made showing the square index of each complement number,
  (number 13 gets its own index)
• such a sequence is made for each aspect of the square
• the lowest order sequence is chosen as the group form of the square

Example square and and its 1-dimensional array:

      2  3 14 22 24
      8 23 19 11  4
     25 20  1  9 10
     13 12 16 18  6
     17  7 15  5 21

 2 3 14 22 24 8 23 19 11 4 25 20 1 9 10 13 12 16 18 6 17 7 15 5 21

The 8 aspects of the square:

      2  3 14 22 24   17 13 25  8  2   21  5 15  7 17   24  4 10  6 21
      8 23 19 11  4    7 12 20 23  3    6 18 16 12 13   22 11  9 18  5
     25 20  1  9 10   15 16  1 19 14   10  9  1 20 25   14 19  1 16 15
     13 12 16 18  6    5 18  9 11 22    4 11 19 23  8    3 23 20 12  7
     17  7 15  5 21   21  6 10  4 24   24 22 14  3  2    2  8 25 13 17

     24 22 14  3  2   21  6 10  4 24   17  7 15  5 21    2  8 25 13 17
      4 11 19 23  8    5 18  9 11 22   13 12 16 18  6    3 23 20 12  7
     10  9  1 20 25   15 16  1 19 14   25 20  1  9 10   14 19  1 16 15
      6 18 16 12 13    7 12 20 23  3    8 23 19 11  4   22 11  9 18  5
     21  5 15  7 17   17 13 25  8  2    2  3 14 22 24   24  4 10  6 21

The corresponding sequences of the pair indexes:

 4  6 16  9  0 18  1 21 22  3 12 19 10 20 17 15  2 14  5 11 13  7  8 24 23
17  1 12 16 24 13 14 21  9  8 18 22  2  5  6 20  3  0 10 23 15  7 11 19  4
 1  0 16 17 11 13 19 10 22  9  7  4 14  5 12 21  2  3 23  6 24 15  8 18 20
20  5 13 17  9  1 14 24 21  4 18 19 22  2  6 16 15  3 10 11  0  8 12 23  7
 4  5 18  8  0  1 22 23  3 16 17 24 14 15 12 13  9 10  2 19 21 20  6  7 11
 5 17 11  9 24  0 23 20 10  3  8  2 22 15 16 13 14  1 19 18  7 21 12  6  4
13 17 18  4  3  5 22 14 15 11 12  9 10  0  7  8 21  1  2 23 24 16  6 19 20
20 18 12  3 17  6  5 23 10 11  8  9  2 22 16 21 14  4  1 24  0 15 13  7 19

The group form of the square is:

 1 0 16 17 11 13 19 10 22 9 7 4 14 5 12 21 2 3 23 6 24 15 8 18 20

Pairs having a number at index 1, 3, 5, 9, 15, 19, 21 or 23 are shown in color.

Group Order

The groups are numbered by size in descending order, i.e., from big to small.
Groups of the same size are numbered in descending order of their group forms.

Groups

There are 101,774,553 groups. The biggest group has 228,960 squares. The 87,430,664 smallest groups each have 2, (complementary), squares.

There are 617 group sizes.

REFERENCES

Heinz, H.D. and J.R. Hendricks "Magic Square Lexicon: Illustrated"
http://recmath.org/Magic%20Squares/Downloads/Lexicon_Sample.pdf,
Magic Square Lexicon: Illustrated, Complementary pair patterns, page 24.