  
  [1X1 [33X[0;0YThe [5XSOTGrps[105X[101X[1X package[133X[101X
  
  [33X[0;0YWith  some  overlaps, the [5XSOTGrps[105X package extends the Small Group Library to
  give access to some more [21Xsmall[121X orders. For example, it constructs a complete
  and  irredundant  list  of isomorphism type representatives of the groups of
  order[133X
  
  [30X    [33X[0;6Ythat factorises into at most four primes;[133X
  
  [30X    [33X[0;6Y[22Xp^4q[122X, for distinct primes [22Xp[122X and [22Xq[122X.[133X
  
  [33X[0;0YThe mathematical background for this package is described in [DEP22].[133X
  
  
  [1X1.1 [33X[0;0YMain functions[133X[101X
  
  [33X[0;0YIn  addition  to  the  functions  described  below, the [5XSOTGrps[105X package also
  extends  the  the  Small Groups Library as provided by the [5XSmallGrp[105X package:
  with  [5XSOTGrps[105X  loaded,  functions  such  as [10XNumberSmallGroups[110X, [10XSmallGroup[110X or
  [10XIdGroup[110X will work for orders support by [5XSOTGrps[105X but not by [5XSmallGrp[105X.[133X
  
  [33X[0;0YNote: for orders support by [5XSOTGrps[105X *and* by [5XSmallGrp[105X, the respective ids as
  produced  by  [10XIdGroup[110X versus [10XIdSOTGroup[110X in general do not agree. In a future
  version we may provided functions to convert between them.[133X
  
  [1X1.1-1 AllSOTGroups[101X
  
  [33X[1;0Y[29X[2XAllSOTGroups[102X( [3Xn[103X[, [3Xfilter[103X] ) [32X function[133X
  
  [33X[0;0Ytakes  in  a  number [3Xn[103X that factorises into at most four primes or is of the
  form   [22Xp^4q[122X  ([22Xp[122X,  [22Xq[122X  are  distinct  primes),  and  returns  a  complete  and
  duplicate-free  list  of  isomorphism class representatives of the groups of
  order  [3Xn[103X.  Solvable  groups  are  using refined polycyclic presentations. By
  default, solvable groups are constructed in the filter [10XIsPcGroup[110X, but if the
  optional   argument  [3Xfilter[103X  is  set  to  [10XIsPcpGroup[110X  then  the  groups  are
  constructed  in  that filter instead. Nonsolvable groups are always returned
  as permutation groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAllSOTGroups(60);[127X[104X
    [4X[28X[ <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,[128X[104X
    [4X[28X  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,[128X[104X
    [4X[28X  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,[128X[104X
    [4X[28X  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,[128X[104X
    [4X[28X  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,[128X[104X
    [4X[28X  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,[128X[104X
    [4X[28X Alt( [ 1 .. 5 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X1.1-2 NumberOfSOTGroups[101X
  
  [33X[1;0Y[29X[2XNumberOfSOTGroups[102X( [3Xn[103X ) [32X function[133X
  
  [33X[0;0Ytakes  in a number [3Xn[103X that factorises into at most four primes or of the form
  [22Xp^4q[122X ([22Xp[122X, [22Xq[122X are distinct primes), and returns the number of isomorphism types
  of groups of order [3Xn[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumberOfSOTGroups(2*3*5*7);[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27XNumberOfSOTGroups(2*3*5*7*11);[127X[104X
    [4X[28XError, Order 2310 is not supported by SOTGrps.[128X[104X
    [4X[28XPlease refer to the SOTGrps documentation for the list of supported orders.[128X[104X
  [4X[32X[104X
  
  [1X1.1-3 SOTGroup[101X
  
  [33X[1;0Y[29X[2XSOTGroup[102X( [3Xn[103X, [3Xi[103X[, [3Xarg[103X] ) [32X function[133X
  
  [33X[0;0Ytakes in a pair of numbers [3Xn, i[103X, where [3Xn[103X factorises into at most four primes
  or  of  the form [22Xp^4q[122X ([22Xp[122X, [22Xq[122X are distinct primes), and returns the [3Xi[103X-th group
  with   respect   to   the  ordering  of  the  list  [10XAllSOTGroups([3Xn[103X[10X)[110X  without
  constructing  all  groups in the list. The option of constructing a PcpGroup
  is available for solvable groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSOTGroup(2*3*5*7, 1);[127X[104X
    [4X[28X<pc group of size 210 with 4 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  the input [3Xi[103X exceeds the number of groups of order [3Xn[103X, an error message is
  returned.[133X
  
  [1X1.1-4 IdSOTGroup[101X
  
  [33X[1;0Y[29X[2XIdSOTGroup[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0Ytakes  in  a group of order determines the SOT library number of [3XG[103X; that is,
  the  function  returns a pair [[3Xn[103X, [3Xi[103X] where [3XG[103X is isomorphic to [10XSOTGroup([3Xn[103X[10X,[3Xi[103X[10X)[110X.
  Note that if the input group is a PcpGroup, this may result in slow runtime,
  as  [10XIdSOTGroup[110X  may  compute the [10XCentre[110X and/or the [10XFittingSubgroup[110X, which is
  slow for PcpGroups.[133X
  
  [1X1.1-5 IsIsomorphicSOTGroups[101X
  
  [33X[1;0Y[29X[2XIsIsomorphicSOTGroups[102X( [3XG[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0Ydetermines  whether  two  groups [3XG[103X, [3XH[103X are isomorphic. It is assumed that the
  input groups are available in the [5XSOTGrps[105X library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=Image(IsomorphismPermGroup(SmallGroup(690,1)));;[127X[104X
    [4X[25Xgap>[125X [27XH:=Image(IsomorphismPcGroup(SmallGroup(690,1)));;[127X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicSOTGroups(G,H);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X1.1-6 IsSOTAvailable[101X
  
  [33X[1;0Y[29X[2XIsSOTAvailable[102X( [3Xn[103X ) [32X function[133X
  
  [33X[0;0Yreturns  [9Xtrue[109X  if the order [3Xn[103X is available in the [5XSOTGrps[105X library, and [9Xfalse[109X
  otherwise.[133X
  
  [1X1.1-7 SOTGroupsInformation[101X
  
  [33X[1;0Y[29X[2XSOTGroupsInformation[102X( [3Xn[103X ) [32X function[133X
  
  [33X[0;0Yprints  information  on  the  groups of the specified order. Since there are
  some  overlaps  between  the  existing  SmallGrps  library  and  the [5XSOTGrps[105X
  library.  In  particular,  [5XSOTGrps[105X  may  construct the groups in a different
  order  and  so  generate a different group ID; we denote such IDs by [9XSOT[109X. If
  the  order  covered  in  [5XSOTGrps[105X  library has no conflicts with the existing
  library, then such a flag is removed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSOTGroupsInformation(2^2*3*19);[127X[104X
    [4X[28X[128X[104X
    [4X[28X  There are 15 groups of order 228.[128X[104X
    [4X[28X[128X[104X
    [4X[28X  The groups of order p^2qr are either solvable or isomorphic to Alt(5).[128X[104X
    [4X[28X  The solvable groups are sorted by their Fitting subgroup.[128X[104X
    [4X[28X     SOT 1 - 2 are the nilpotent groups.[128X[104X
    [4X[28X     SOT 3 has Fitting subgroup of order 57.[128X[104X
    [4X[28X     SOT 4 - 7 have Fitting subgroup of order 76.[128X[104X
    [4X[28X     SOT 8 - 9 have Fitting subgroup of order 38.[128X[104X
    [4X[28X     SOT 10 - 15 have Fitting subgroup of order 114.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSOTGroupsInformation(2662);[127X[104X
    [4X[28X[128X[104X
    [4X[28X There are 15 groups of order 2662.[128X[104X
    [4X[28X[128X[104X
    [4X[28X The groups of order p^3q are solvable by Burnside's pq-Theorem.[128X[104X
    [4X[28X These groups are sorted by their Sylow subgroups.[128X[104X
    [4X[28X   1 - 3 are abelian.[128X[104X
    [4X[28X   4 - 5 are nonabelian nilpotent and have a normal Sylow 11-subgroup and a[128X[104X
    [4X[28X       normal Sylow 2-subgroup.[128X[104X
    [4X[28X   6 is non-nilpotent and has a normal Sylow 2-subgroup with Sylow[128X[104X
    [4X[28X      11-subgroup [ 1331, 1 ].[128X[104X
    [4X[28X   7 - 9 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow[128X[104X
    [4X[28X      11-subgroup [ 1331, 2 ].[128X[104X
    [4X[28X   10 - 12 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow[128X[104X
    [4X[28X      11-subgroup [ 1331, 5 ].[128X[104X
    [4X[28X   13 - 14 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow[128X[104X
    [4X[28X      11-subgroup [ 1331, 3 ].[128X[104X
    [4X[28X   15 is non-nilpotent and has a normal Sylow 2-subgroup with Sylow[128X[104X
    [4X[28X     11-subgroup [ 1331, 4 ].[128X[104X
  [4X[32X[104X
  
