  
  [1X7 [33X[0;0YExamples with Runtimes[133X[101X
  
  [33X[0;0YIn  this  chapter  we  outline  some examples of applications of the methods
  described  above.  The  examples  are  meant to give an idea of the possible
  applications of the package. Thus we included runtimes for all examples, but
  omitted  the output in some cases, since it would be too long to be printed.
  The runtimes have been obtained on a 400 MHz PC running under Linux.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XConstructAllGroups( 60 );; time;[127X[104X
    [4X[28X4080[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the following examples we observe that the restriction to certain groups
  is  often  helpful.  Note  that  nilpotent groups can often be obtained as a
  direct  product  of  [22Xp[122X-groups which, in turn, might better be constructed by
  [22Xp[122X-group generation methods.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFrattiniExtensionMethod( 5^3 * 7 * 31, true );;[127X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X13670[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xflags := rec( nonnilpot := true );;[127X[104X
    [4X[25Xgap>[125X [27XFrattiniExtensionMethod( 5^3 * 7 * 31, flags, true );;[127X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X8400[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xflags := rec( nonsupsol := true );;[127X[104X
    [4X[25Xgap>[125X [27XFrattiniExtensionMethod( 5^3 * 7 * 31, flags, true );;[127X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X3640[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xflags := rec( nonpnorm := [31] );;[127X[104X
    [4X[25Xgap>[125X [27XFrattiniExtensionMethod( 5^3 * 7 * 31, flags, true );;[127X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X1740[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  consider  groups  of an order whose factorisation contains a large
  prime.  Note  that  the  Small  Groups  library contains a generic method to
  construct  the  groups  whose order is the product of at most 3 primes. This
  method  is used in [10XConstructAllGroups[110X which is therefore much more efficient
  in the next example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFrattiniExtensionMethod( 10007 * 2, true );[127X[104X
    [4X[28X[ <pc group of size 20014 with 2 generators>, [128X[104X
    [4X[28X  <pc group of size 20014 with 2 generators> ][128X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X87950[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xflags := rec( nonnilpot := true );;[127X[104X
    [4X[25Xgap>[125X [27XFrattiniExtensionMethod( 10007 * 2, flags, true  );[127X[104X
    [4X[28X[ <pc group of size 20014 with 2 generators> ][128X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X48950[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XConstructAllGroups( 10007 * 2 );[127X[104X
    [4X[28X[ <pc group of size 20014 with 2 generators>, [128X[104X
    [4X[28X  <pc group of size 20014 with 2 generators> ][128X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X30[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally we consider an order which factorises into seven primes and contains
  a moderately large prime power. Note that there are 943 non-nilpotent groups
  of  order [22X288 = 2^5 ⋅ 3^2[122X while there are only 90 such groups without normal
  Sylow subgroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xflags := rec( nonnilpot := true );;[127X[104X
    [4X[25Xgap>[125X [27XFrattiniExtensionMethod( 2^5 * 3^2, flags, true );;[127X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X656630[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xflags := rec( nonpnorm := [2,3] );;[127X[104X
    [4X[25Xgap>[125X [27XFrattiniExtensionMethod( 2^5 * 3^2, flags, true );;[127X[104X
    [4X[25Xgap>[125X [27Xtime;[127X[104X
    [4X[28X58180[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y [133X
  
