This document serves as a guide to the various files relating to Suz(512).

1) The file Suz512.txt contains all of the basic programs that are used to generate the data used in the other programs.
   In particular, it includes code to generate all of the possible elements of order 37 that can be extended to an element
   of order 481, with a good power to an element of order 13. It also includes code to check that the elements of order
   481 are blueprints. It is checked for every element of order 481 that is found is a blueprint.

2) The file Suz512NoneCode.txt includes the code to find all sets of composition factors conspicuous for an element of
   order 481 that power to given elements of orders 37 and 13, found by Suz512.txt. This only finds the sets of composition
   factors that have no 64-dimensional factors. You have to run a command like
   
   magma
   jj:=1;
   load "Subgroups/E8/Suz512/E8RepsOrder13Case4.txt";
   load "Subgroups/E8/Suz512/Suz512NoneCode.txt";
   
   The quantity jj takes values from 1 to 19. The particular Case one loads is given in Suz512.txt. When accounting for
   a few extra conditions and symmetries, the possibilities for jj and the Case are given in this double array:
   
   [[8,10,12,14],[8,10,12,14],[8,10,12,14],[10,12,14],[8,10,12,14],[10,12,14],[8,14],
   [8,10,12,14],[8,10,12,14],[15],[8,15],[8],[8,10,12,14],[14],[14],[8],[],[],[]];

   What this means that is if jj=1, then we have Cases 8, 10, 12 and 14, and so on.

   One finds conspicuous sets of composition factors for the values: jj=7, Case 14; jj=8, Case 12; jj=9, Case 8; jj=15, 
   Case 14; jj=16, Case 8. The rest yield no solutions. These are the five conspicuous sets of factors given in the text.

   If the Case is 8 then one run of this program takes about an hour, but for the others it takes about 4 hours per run.

3) For one 64-dimensional factor, there is a difference depending on whether the factor restricts to Suz(8) as 64 or not.
   In the former case, one always has Case 16/17/18, and this takes a lot longer to run than the other possibilities for
   the restriction to Suz(8). This is a separate file, Suz512One16Code.txt. This takes a single parameter, jj, with values
   between 1 and 81. This time run a command such as 

   magma
   jj:=1;
   load "Subgroups/E8/Suz512/Suz512One16Code.txt";

   This actually finds all sets of factors conspicuous for an element of order 481, whose 64-dimensional factors are numbers
   47, 50 and 55. Running this code for a given value of jj generally takes roughly 15 hours or so. We don't need to do this
   for many though, because the answers can be subsumed in code that has to be run for the two 64-dim case later.

   See the results file for the particular answers from this. This is where the single case of one 64 appears, with the code

   screen -DR suz1case16no48
   magma
   jj:=48;
   load "Subgroups/E8/Suz512/Suz512OneCode.txt";

   which returns
   [[12,4,6,1,0,0,4,4,0,0,2,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0]]
   [[ 308, 529, 364, 699, 550, 699, 502, 747 ]]
   [ 15869435 ]

4) The files Suz512One4Code.txt, Suz512One5Code.txt and Suz512One8Code.txt, when run, check that there are no conspicuous
   sets of composition factors with a single 64-dimensional composition factor and whose restriction to Suz(8) is from the
   cases 4, 5 and 8 given in the file Suz512.txt.

   The first of these finds a solution but it has no 64s and was previously found (up to field automorphism) as jj=9, Case 8:
   [16 0 8 1 4 0 0 4 8 1 0 0 0 0 0 0 0 0 0 2 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0]

   The second of these finds four solutions:
   [12 4 1 1 0 0 5 4 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0]
   [12 0 1 1 4 0 4 4 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0]
   [16 8 1 4 0 0 4 8 1 0 0 2 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0]
   [16 8 1 0 8 1 4 0 0 4 0 0 4 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
   The last two of these are (up to field automorphism) the same as jj=9, Case 8. The first two are new, but have two
   64-dimensional factors.

   The third of these finds four solutions:
   [16 1 4 0 0 4 8 1 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
   [16 0 4 8 1 0 8 1 4 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0]
   [16 1 0 8 1 4 0 0 4 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0]
   [12 0 1 4 1 0 4 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0]
   The first three of these are again the same as jj=9, Case 8. The last of these is new, but has two 64-dimensional
   factors.

5) The files Suz512One10Code.txt, Suz512One12Code.txt and Suz512One14Code.txt take a single parameter jj, which runs
   between 1 and 21 for the first case, 1 and 19 for the second case, and 1 and 20 for the third case. We find quite
   a few solutions, all having either no or two 64-dimensional modules, so will appear later on. We do not give the
   precise cases that work, leaving that to Suz512results.txt, as there are lots of cases that turn up solutions,
   although none with exactly one 64-dimensional factor. Again, many of these cases don't need to be run, because it
   turns out that they will be subsumed in the next two files.

6) The files Suz512Two16Code.txt is the one that looks for all solutions with exactly two 64-dimensional summands, at
   least one of which restricts irreducibly to Suz(8). Of course, because it just searches for a rational solution to
   a system of linear equations, it also picks up those solutions with one or no 64-dimensional factors, given the
   same traces for elements of orders 37 and 13. Because there is a 64 in the restriction ot Suz(8), this uses the
   trace of a 13 coming from Cases 16/17/18/19, so this file subsumes most of the cases of Suz512OneCode.txt. However,
   some of the possibilities for a 37 there do not appear here, so that is why some of those (but not many) still
   need to be run.

7) The files Suz512Two4Code.txt, Suz512Two5Code.txt, Suz512Two8Code.txt, Suz512Two10Code.txt, Suz512Two12Code.txt and
   Suz512Two14Code.txt are the last files that need to be run. The first three can just be run, the last three take a
   parameter jjj, which runs from 1 to 7 in the first two cases, and 1 to 8 in the last. These look for all solutions
   with two 64-dimensional summands, but whose restriction to Suz(8) has no 64s.

8) The file Suz512PossElts.txt contains representatives of the various conjugacy classes of elements of order 37 that
   are used in this proof. Noting that there's only around a hundred conjugacy classes in total, it could be possible
   to produce a faster proof of this, working one element of order 37 at a time and considering first those preimages
   or order 481 that have a rational solution to the eigenspace problem, then work from there to nail down which do.
   This requires one to know a priori that there are few classes in play, which of course was not known at the start
   of the proof. I thought there was limited usefulness in spending a large amount of time rewriting the programs
   to take advantage of this fact.

   In this file we also include the results of Soln and ThingsThatWork from Suz512 when ToRem has size 1. These are
   important for computing the input data for the code in 3), 4) and 5) above. We also give the code needed to
   produce the data for the files in 6) and 7) above.

9) The file Suz512results.txt collates the results of all programs that give solutions, together with the precise
   code needed to produce the solutions. The elements here are fed into Suz512.txt to prove that every case is a
   blueprint.

10) The file Suz512AllCommands gives a complete list of all commands that must be run in order to generate the whole
   of the results above, and to confirm there are no more. Because there is significant overlap between the possible
   elements of order 37 that appear on the lists Soln for the various possibilities for ToRem, we can cut down on
   the number of cases considerably. When running it myself, I ran all cases rather than removing the duplication,
   as I felt that the saving to CPU time was not worth the cost to my time of recording which ones do not need to be
   run. In addition, I felt that running all gave double security that all cases had indeed been found.

11) The files E8RepsOrder13Case4 and so on give all elements of order 13 in a torus with the eigenvalues from Case 4.
   Code to generate these files is given in Suz512.txt. Note that one only needs to compute the files for elements
   ending in 0 and 1, because then all others are obtained by taking powers of those elements.
