  
  [1X2 [33X[0;0YAccessing the data[133X[101X
  
  [33X[0;0YWhen  this package is loaded, the groups of order [22X3^8[122X and [22Xp^7[122X for primes [22Xp >
  11[122X  become additionally available via the Small Groups Library. As a result,
  all  groups  of order [22Xp^n[122X with [22Xp=2[122X and [22Xn ≤ 9[122X, with [22Xp=3[122X and [22Xn ≤ 8[122X, and with [22Xp[122X
  an  arbitrary  prime  and  [22Xn  ≤  7[122X  are  then available via the Small Groups
  Library.  The  corresponding  information  can be obtained via the following
  functions.[133X
  
  
  [1X2.1 [33X[0;0YFunctions for accessing the data[133X[101X
  
  [1X2.1-1 SmallGroup[101X
  
  [33X[1;0Y[29X[2XSmallGroup[102X( [3Xsize[103X, [3Xnumber[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumberSmallGroups[102X( [3Xsize[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSmallGroupsInformation[102X( [3Xsize[103X ) [32X function[133X
  
  [33X[0;0YSee  [14X'smallgrp: The Small Groups Library'[114X for background on these functions.
  Note  that  there is no [10XIdGroup[110X function available for this extension of the
  Small Groups Library.[133X
  
  [33X[0;0YWARNING:  The  user should be aware that there are 1,396,077 groups of order
  [22X3^8[122X, 1,600,573 groups of order [22X13^7[122X, and 5,546,909 groups of order [22X17^7[122X. For
  general  [22Xp[122X, the number of groups of order [22Xp^7[122X is a PORC polynomial in [22Xp[122X with
  leading  term [22X3p^5[122X. Furthermore, as the prime [22Xp[122X increases, the time taken to
  generate  a  complete  list  of  the  groups  of  order  [22Xp^7[122X  grows rapidly.
  Experimentally,  the  time  seems  to  be proportional to [22Xp^6.2[122X. For [22Xp=13[122X it
  takes  several  hours  to  generate  the complete list. For primes [22Xp≤ 11[122X the
  groups  are  precomputed, and their SmallGroup codes are stored in the Small
  Groups Library. For primes [22Xp>11[122X the Lie rings have to be generated from 4773
  parametrized presentations in the LiePRing database, and then converted into
  groups  using  the  Baker-Campbell-Hausdorff  formula.  A  complete  list of
  power-commutator presentations for the groups of order [22X13^7[122X takes over 11 GB
  of memory.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumberSmallGroups(3^8);[127X[104X
    [4X[28X1396077[128X[104X
    [4X[25Xgap>[125X [27XSmallGroup(3^8, 1000000);[127X[104X
    [4X[28X<pc group of size 6561 with 8 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  groups  of  order  [22Xp^7[122X  for [22Xp>11[122X are only available if the [5XLiePRing[105X and
  [5XLieRing[105X  packages  are loaded. They are constructed in batches, so the first
  request for such a group reports what it is doing and may take a while:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumberSmallGroups(17^7);[127X[104X
    [4X[28X5546909[128X[104X
    [4X[25Xgap>[125X [27XNumberSmallGroups(101^7);[127X[104X
    [4X[28X32826263845[128X[104X
    [4X[25Xgap>[125X [27XSmallGroup(17^7, 5000);[127X[104X
    [4X[28Xconstructing a batch of 1156 groups ... this may take a while[128X[104X
    [4X[28X<pc group of size 410338673 with 7 generators>[128X[104X
  [4X[32X[104X
  
