Text from Francis Gaspalou emails:

It should be interesting also to identify the sets of squares which are
isomorphic to your set of 21,446.

At a first look, when applying the group G192, there are 192/8=24 isomorphic
sets (and then a grand total of 24*21,446 squares). Each isomorphic set can
be defined by 7*4=28 relations of definition which are the transformed of
the bent diagonals relations.

For example, for the set coming from the transformation (1324657)all, the
relations of definition are:

Each isomorphic set has also the same features, i.e. D4=25, A1+A4+A7=75,

Sorry, I wrote too rapidly "Each isomorphic set has also the same features, i.e. D4=25, A1+A4+A7=75, etc.".

In fact, for 8 isomorphic sets out of 24, we have A1+A4+A7=75, .. and A1+D1+G1=75,. 
for 8 other sets, A2+A4+A6=75,.and B1+D1+F1=75,.
and for 8 other sets, A3+A4+A5=75,.and C1+D1+E1=75,. 

I demonstrated your properties (D4=25, A1+A4+A7=75, etc) from the 7*4=28 relations of definition of the bent diagonals.

Some details:
Exactly, I searched to reduce the number of a set of 6*6=36 initial parameters when writing the magic conditions
and the bent diagonals conditions. It was a tedious task made by hand (about 13 pages of calculus).

In particular, I found 4 relations at the end:
It is a system of 4 equations for 4 unknown parameters and the only solutions are
A2+A3+A5+A6=100, B2+B3+B5+B6=100, A3+A5+B4=75, D4=25.

You see that there are supplementary properties like A3+A5+B4=75.

With the set of the reduced parameters, it is possible to build a backtracking program of enumeration.
I think that this backtracking program with optimized parameters should give the solutions in a
reasonable time but I don’t plan to make such a program of verification.

With my analysis, I found that the number of parameters is 15, for example
A1, A2, A3, A4, A5
B1, B2, B3, B5
C1, C2
D1, D2, D3, D5
(all the remaining cells can be calculated in function of these parameters: I have the formulae).

If you want, you can add the table in attachment ... .

From this table, we can deduce many others relations like
But these relations don’t reduce more the number of parameters (15).



Text from Miguel Angel Amela emails:

Effectively, the 7x7 bent diagonal 4 pan-way magic squares with the properties
White-Gaspalou they can be formulated with 15 independent variables (parameters).  
Here I send a source code written in Quick Basic.

Confirmed: the number of solutions without rotations nor reflections of 7x7 bent diagonals 4 pan-way magic squares with a1 = 22 is 141.
In a PC Microsoft Windows XP Professional, Version 2002. Service Pack 3; AMD Athlon (tm) II X2 245, 2.91 GHz; RAM 1,75 GB and
compiling with QB64 the running time was 2316 seconds (38 minutes).

In the new version, the source code has been very optimized by using other sequence of variables and eliminating unnecessary checkups.

If Harry translates the source code to the C-language, is of my interest to know the running time for a1 = 22.

Here I send the source code as .BAS and .TXT and the screen capture of the result.

The property of the subsets of five entries that add 125 allows a sequence of variables a lot of more optimum ... the reduction of the
running time is really impressive !!!

Now and for a1 = 22, the 141 squares are obtained in 267 seconds (4,8 minutes).

Here I send 7x7BDM-H.bas, 7x7BDM-H.txt and the screen capture of the result.

Here is 7x7BDM-H.txt.

The C translation uses a boolean array to keep track of number status, (in use or free). The running time for a1 = 22, (141 squares), on a 3.0 GHz PC is 25 seconds. The running time for a1 = 1 .. 49, (21,446 squares), is 2:11:40.