  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1 [33X[0;0YPhilosophy[133X[101X
  
  [33X[0;0YThe  package  [5XClassicalMaximals[105X provides functionality for computing maximal
  subgroups  of  finite  classical  quasisimple groups in their natural matrix
  representations (as returned by the [5XGAP[105X functions [10XSL[110X, [10XSU[110X, [10XSp[110X and [10XOmega[110X). Its
  primary  purpose  is  to  return  a list of representatives of the conjugacy
  classes  of  maximal  subgroups  of  a  given classical group defined over a
  finite field.[133X
  
  [33X[0;0YThe  implementation  follows  the  classification given in the book by Bray,
  Holt  and  Roney-Dougal  [BHR13].  The maximal subgroups of finite classical
  groups  are  divided,  according to Aschbacher's theorem, into nine classes.
  The  subgroups  in  the  first  eight  classes  are referred to as [13Xgeometric
  subgroups[113X.  These  admit a uniform description, which is developed in detail
  in   the  literature  (see,  for  example,  [KL90]).  In  particular,  their
  construction  can  be implemented in a systematic way, which is done in this
  package  by  directly  implementing  algorithms  from  the papers [HR05] and
  [HR10].[133X
  
  [33X[0;0YIn  contrast,  subgroups  in  the  ninth  class [23X{\cal S}[123X do not admit such a
  uniform  description  and  must be determined separately for each dimension.
  The  lists  given  in  [HM01]  and [Lüb01] contain, in principle, sufficient
  information  to  determine  these  subgroups  up  to dimension 250. However,
  explicit  computations have so far only been carried out up to dimension 17,
  [BHR13]  does  so  up  to dimension 12 and for these dimensions our returned
  lists  should  therefore  be  complete.  For  higher  dimensions  we give no
  completeness  guarantee;  in particular, all subgroups in class [23X{\cal S}[123X are
  missing in these cases.[133X
  
  [33X[0;0YFor  the maximal subgroups in class [23X{\cal S}[123X, our code is in several parts —
  and   in   some   cases   almost   entirely   —  a  translation  of  Magma's
  [10XClassicalMaximals[110X  function  (written by Derek Holt and Colva Roney-Dougal).
  The  package  therefore benefits from the extensive prior work done in Magma
  while attempting to adapt it to the [5XGAP[105X system.[133X
  
  [33X[0;0YThe   central   entry   point   of   [5XClassicalMaximals[105X   is   the   function
  [2XClassicalMaximalsGeneric[102X ([14X3.1-1[114X). This function takes parameters that define
  a  classical  group, together with options controlling the type of subgroups
  to  compute,  and  returns  representatives  of  maximal subgroups organised
  according  to  the  classification  arising  from  Aschbacher's  theorem. An
  illustrive call looks as follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XClassicalMaximalsGeneric("L", 3, 4, [1..8]);[127X[104X
    [4X[28X[ <matrix group of size 2880 with 5 generators>,[128X[104X
    [4X[28X  <matrix group of size 2880 with 5 generators>,[128X[104X
    [4X[28X  <matrix group of size 504 with 3 generators>,[128X[104X
    [4X[28X  <matrix group of size 504 with 3 generators>,[128X[104X
    [4X[28X  <matrix group of size 504 with 3 generators>,[128X[104X
    [4X[28X  <matrix group of size 216 with 3 generators> ][128X[104X
  [4X[32X[104X
  
  
  [1X1.2 [33X[0;0YOverview over this manual[133X[101X
  
  [33X[0;0YChapter  [14X2[114X provides a brief theoretical introduction to classical groups and
  Aschbacher's theorem, focusing on the concepts essential for this package.[133X
  
  [33X[0;0YChapter  [14X3[114X documents the package's core functionality, specifically the main
  function [2XClassicalMaximalsGeneric[102X ([14X3.1-1[114X).[133X
  
  [33X[0;0YChapter  [14X4[114X  contains  examples  illustrating  the  typical  use  of the main
  function.[133X
  
  [33X[0;0YChapter  [14X5[114X  lists  utility  functions  that,  while  auxiliary,  may  be  of
  independent interest to the user.[133X
  
  
  [1X1.3 [33X[0;0YFeedback and Support[133X[101X
  
  [33X[0;0YReport  bugs,  questions  and issues on the [5XClassicalMaximals[105X issue tracker:
  [7Xhttps://github.com/gap-packages/ClassicalMaximals/issues[107X[133X
  
  
  [1X1.4 [33X[0;0YFunding[133X[101X
  
  [33X[0;0YThe  development  of  this  [5XGAP[105X  package is supported by the German Research
  Foundation   (DFG)   within   the  Collaborative  Research  Center  TRR  195
  ([7Xhttps://www.computeralgebra.de/sfb/[107X).[133X
  
