Frénicle standard form

A magic square is in Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions apply:

Frénicle's published book of 1693 described all the 880 essentially different order-4 magic squares.

This standard form was devised since a magic square remains "essentially similar" if it is rotated or transposed, or flipped so that the order of rows is reversed — there exists 8 different magic squares sharing one standard form. For example, the following magic squares are all essentially similar, with only the final square being in Frénicle standard form:

For each group of magic squares one might identify the corresponding group of automorphisms, the group of transformations preserving the special properties of this group of magic squares. This way one can identify the number of different magic square classes.

From the perspective of Galois theory the most-perfect magic squares are not distinguishable. This means that the number of elements in the associated Galois group is 1. Please compare   Number of essentially different most-perfect pandiagonal magic squares of order 4n. with   The simplest sequence of positive numbers: the all 1's sequence.

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