\chapter{Code: Finite subgroups of {$\generalLinear(n,\integers)$}}



\section{Code}

Included in the auxiliary documents with this thesis is a digital document titled \fontComputer{fsGLnZwH1.txt}. This document is code, written in Magma, which does the following:
\begin{enumerate}
\item It first produces a sequence, stratified by rank, with exactly one representative for each $\integers$-conjugacy class of finite subgroup of $\generalLinear(n,\integers)$, from rank $n = 1$ to \fontComputer{nMax}. (The parameter \fontComputer{nMax}, defined at the beginning of the code, is set equal to $6$ and adjustable.)
\item It then computes the first cohomology group for each subgroup.
\end{enumerate}





\newpage

\section{Sample output}

{\setstretch{1.0}\footnotesize
\begin{verbatim}
**************************
*                        *
*   Summary of results   *
*                        *
**************************

   | #(Z-conj classes fin subs) | ...of which, among the subs found,... 
   |             |              |            |          | #(not R-irred 
 n |  expected   |    found     | #(R-irred) | #(built) |   not built)  
---+-------------+--------------+------------+----------+---------------
 1 |       2     |       2      =       2    +      0   +        0
 2 |      13     |      13      =       7    +      6   +        0
 3 |      73     |      73      =      15    +     58   +        0
 4 |     710     |     710      =     134    +    567   +        9
 5 |    6079     |    6079      =      79    +   6000   +        0
 6 |   85308     |   85308      =     893    +  84399   +       16

Saving Magma state to "fsGLnZ_1_6.ws"

*********************************
*                               *
*   Rank 1 : Finite subgroups   *
*                               *
*********************************

Set of H^1 :
{*
[],
[ 2 ]
*}

Set of H^1 (cyclic) :
{*
[],
[ 2 ]
*}

Set of H^1 (cyclic) = Set of H^1? true

Set of H^1 (K3-admissible) :
{*
[]
*}

Set of H^1 (K3-admissible and cyclic) :
{*
[]
*}

Set of H^1 (K3-adm, cyc) = Set of H^1 (K3-adm)? true


*********************************
*                               *
*   Rank 2 : Finite subgroups   *
*                               *
*********************************

Set of H^1 :
{*
[]^^5,
[ 2 ]^^4,
[ 2, 2 ]^^2,
[ 3 ]^^2
*}

Set of H^1 (cyclic) :
{*
[]^^3,
[ 2 ]^^2,
[ 2, 2 ],
[ 3 ]
*}

Set of H^1 (cyclic) = Set of H^1? true

Set of H^1 (K3-admissible) :
{*
[]^^2,
[ 2 ]
*}

Set of H^1 (K3-admissible and cyclic) :
{*
[]^^2,
[ 2 ]
*}

Set of H^1 (K3-adm, cyc) = Set of H^1 (K3-adm)? true


*********************************
*                               *
*   Rank 3 : Finite subgroups   *
*                               *
*********************************

Set of H^1 :
{*
[]^^15,
[ 2 ]^^31,
[ 2, 2 ]^^13,
[ 2, 2, 2 ]^^4,
[ 3 ]^^2,
[ 4 ]^^5,
[ 6 ]^^3
*}

Set of H^1 (cyclic) :
{*
[]^^5,
[ 2 ]^^5,
[ 2, 2 ]^^2,
[ 2, 2, 2 ],
[ 3 ],
[ 4 ],
[ 6 ]
*}

Set of H^1 (cyclic) = Set of H^1? true

Set of H^1 (K3-admissible) :
{*
[]^^10,
[ 2 ]^^7,
[ 2, 2 ]^^2,
[ 3 ]^^2
*}

Set of H^1 (K3-admissible and cyclic) :
{*
[]^^5,
[ 2 ]^^3,
[ 2, 2 ],
[ 3 ]
*}

Set of H^1 (K3-adm, cyc) = Set of H^1 (K3-adm)? true


*********************************
*                               *
*   Rank 4 : Finite subgroups   *
*                               *
*********************************

Set of H^1 :
{*
[]^^172,
[ 2 ]^^223,
[ 2, 2 ]^^175,
[ 2, 2, 2 ]^^42,
[ 2, 2, 2, 2 ]^^8,
[ 2, 4 ]^^11,
[ 2, 6 ]^^7,
[ 3 ]^^33,
[ 3, 3 ]^^6,
[ 4 ]^^13,
[ 5 ]^^5,
[ 6 ]^^15
*}

Set of H^1 (cyclic) :
{*
[]^^13,
[ 2 ]^^12,
[ 2, 2 ]^^6,
[ 2, 2, 2 ]^^2,
[ 2, 2, 2, 2 ],
[ 2, 4 ],
[ 2, 6 ],
[ 3 ]^^4,
[ 3, 3 ],
[ 4 ],
[ 5 ],
[ 6 ]^^2
*}

Set of H^1 (cyclic) = Set of H^1? true

Set of H^1 (K3-admissible) :
{*
[]^^49,
[ 2 ]^^67,
[ 2, 2 ]^^23,
[ 2, 2, 2 ]^^4,
[ 3 ]^^5,
[ 4 ]^^5,
[ 6 ]^^3
*}

Set of H^1 (K3-admissible and cyclic) :
{*
[]^^10,
[ 2 ]^^10,
[ 2, 2 ]^^3,
[ 2, 2, 2 ],
[ 3 ]^^2,
[ 4 ],
[ 6 ]
*}

Set of H^1 (K3-adm, cyc) = Set of H^1 (K3-adm)? true


*********************************
*                               *
*   Rank 5 : Finite subgroups   *
*                               *
*********************************

Set of H^1 :
{*
[]^^497,
[ 2 ]^^1910,
[ 2, 2 ]^^1892,
[ 2, 2, 2 ]^^811,
[ 2, 2, 2, 2 ]^^135,
[ 2, 2, 2, 2, 2 ]^^16,
[ 2, 2, 4 ]^^32,
[ 2, 2, 6 ]^^16,
[ 2, 4 ]^^141,
[ 2, 6 ]^^63,
[ 3 ]^^86,
[ 3, 3 ]^^6,
[ 3, 6 ]^^11,
[ 4 ]^^228,
[ 5 ]^^5,
[ 6 ]^^204,
[ 10 ]^^8,
[ 12 ]^^18
*}

Set of H^1 (cyclic) :
{*
[]^^20,
[ 2 ]^^24,
[ 2, 2 ]^^14,
[ 2, 2, 2 ]^^6,
[ 2, 2, 2, 2 ]^^2,
[ 2, 2, 2, 2, 2 ],
[ 2, 2, 4 ],
[ 2, 2, 6 ],
[ 2, 4 ]^^2,
[ 2, 6 ]^^2,
[ 3 ]^^6,
[ 3, 3 ],
[ 3, 6 ],
[ 4 ]^^5,
[ 5 ],
[ 6 ]^^7,
[ 10 ],
[ 12 ]
*}

Set of H^1 (cyclic) = Set of H^1? true

Set of H^1 (K3-admissible) :
{*
[]^^395,
[ 2 ]^^648,
[ 2, 2 ]^^362,
[ 2, 2, 2 ]^^73,
[ 2, 2, 2, 2 ]^^8,
[ 2, 4 ]^^11,
[ 2, 6 ]^^7,
[ 3 ]^^56,
[ 3, 3 ]^^6,
[ 4 ]^^32,
[ 5 ]^^5,
[ 6 ]^^27
*}

Set of H^1 (K3-admissible and cyclic) :
{*
[]^^20,
[ 2 ]^^19,
[ 2, 2 ]^^11,
[ 2, 2, 2 ]^^3,
[ 2, 2, 2, 2 ],
[ 2, 4 ],
[ 2, 6 ],
[ 3 ]^^6,
[ 3, 3 ],
[ 4 ]^^2,
[ 5 ],
[ 6 ]^^3
*}

Set of H^1 (K3-adm, cyc) = Set of H^1 (K3-adm)? true


*********************************
*                               *
*   Rank 6 : Finite subgroups   *
*                               *
*********************************

Set of H^1 :
{*
[]^^5799,
[ 2 ]^^18950,
[ 2, 2 ]^^29131,
[ 2, 2, 2 ]^^16854,
[ 2, 2, 2, 2 ]^^4116,
[ 2, 2, 2, 2, 2 ]^^460,
[ 2, 2, 2, 2, 2, 2 ]^^36,
[ 2, 2, 2, 4 ]^^90,
[ 2, 2, 2, 6 ]^^38,
[ 2, 2, 4 ]^^762,
[ 2, 2, 6 ]^^256,
[ 2, 4 ]^^2502,
[ 2, 6 ]^^1230,
[ 2, 10 ]^^19,
[ 2, 12 ]^^55,
[ 3 ]^^1404,
[ 3, 3 ]^^159,
[ 3, 3, 3 ]^^17,
[ 3, 6 ]^^64,
[ 4 ]^^1469,
[ 4, 4 ]^^50,
[ 5 ]^^43,
[ 6 ]^^1652,
[ 6, 6 ]^^29,
[ 7 ]^^7,
[ 10 ]^^42,
[ 12 ]^^61,
[ 15 ]^^13
*}

Set of H^1 (cyclic) :
{*
[]^^47,
[ 2 ]^^55,
[ 2, 2 ]^^33,
[ 2, 2, 2 ]^^15,
[ 2, 2, 2, 2 ]^^6,
[ 2, 2, 2, 2, 2 ]^^2,
[ 2, 2, 2, 2, 2, 2 ],
[ 2, 2, 2, 4 ],
[ 2, 2, 2, 6 ],
[ 2, 2, 4 ]^^2,
[ 2, 2, 6 ]^^2,
[ 2, 4 ]^^6,
[ 2, 6 ]^^8,
[ 2, 10 ],
[ 2, 12 ],
[ 3 ]^^19,
[ 3, 3 ]^^4,
[ 3, 3, 3 ],
[ 3, 6 ]^^2,
[ 4 ]^^8,
[ 4, 4 ],
[ 5 ]^^3,
[ 6 ]^^15,
[ 6, 6 ],
[ 7 ],
[ 10 ]^^2,
[ 12 ],
[ 15 ]
*}

Set of H^1 (cyclic) = Set of H^1? true

Set of H^1 (K3-admissible) :
{*
[]^^2370,
[ 2 ]^^7178,
[ 2, 2 ]^^6067,
[ 2, 2, 2 ]^^1817,
[ 2, 2, 2, 2 ]^^235,
[ 2, 2, 2, 2, 2 ]^^16,
[ 2, 2, 4 ]^^32,
[ 2, 2, 6 ]^^16,
[ 2, 4 ]^^250,
[ 2, 6 ]^^111,
[ 3 ]^^310,
[ 3, 3 ]^^17,
[ 3, 6 ]^^11,
[ 4 ]^^465,
[ 5 ]^^13,
[ 6 ]^^394,
[ 10 ]^^8,
[ 12 ]^^18
*}

Set of H^1 (K3-admissible and cyclic) :
{*
[]^^41,
[ 2 ]^^50,
[ 2, 2 ]^^24,
[ 2, 2, 2 ]^^11,
[ 2, 2, 2, 2 ]^^3,
[ 2, 2, 2, 2, 2 ],
[ 2, 2, 4 ],
[ 2, 2, 6 ],
[ 2, 4 ]^^3,
[ 2, 6 ]^^3,
[ 3 ]^^13,
[ 3, 3 ]^^2,
[ 3, 6 ],
[ 4 ]^^7,
[ 5 ]^^2,
[ 6 ]^^12,
[ 10 ],
[ 12 ]
*}

Set of H^1 (K3-adm, cyc) = Set of H^1 (K3-adm)? true
\end{verbatim}
}% End setstretchCode.