Quelle examples.tex Sprache: Latech

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%W examples.tex           GrpConst documentation             Bettina Eick
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\Chapter{Examples with Runtimes}

\index{Examples with Runtimes}

In 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.

\beginexample
gap> ConstructAllGroups( 60 );; time;
4080
\endexample

In the following examples we observe that the restriction to
certain groups is often helpful. Note that nilpotent groups
can often be obtained as direct product of $p$-groups which,
in turn, might better be constructed by $p$-group generation
methods.

\beginexample
gap> FrattiniExtensionMethod( 5^3 * 7 * 31, true );;
gap> time;
13670

gap> flags := rec( nonnilpot := true );;
gap> FrattiniExtensionMethod( 5^3 * 7 * 31, flags, true );;
gap> time;
8400

gap> flags := rec( nonsupsol := true );;
gap> FrattiniExtensionMethod( 5^3 * 7 * 31, flags, true );;
gap> time;
3640

gap> flags := rec( nonpnorm := [31] );;
gap> FrattiniExtensionMethod( 5^3 * 7 * 31, flags, true );;
gap> time;
1740
\endexample

Next we consider groups of an order whose factorision 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 `ConstructAllGroups'
which is therefore much more efficient in the next example.

\beginexample
gap> FrattiniExtensionMethod( 10007 * 2, true );
[ <pc group of size 20014 with 2 generators>,
  <pc group of size 20014 with 2 generators> ]
gap> time;
87950

gap> flags := rec( nonnilpot := true );;
gap> FrattiniExtensionMethod( 10007 * 2, flags, true  );
[ <pc group of size 20014 with 2 generators> ]
gap> time;
48950

gap> ConstructAllGroups( 10007 * 2 );
[ <pc group of size 20014 with 2 generators>,
  <pc group of size 20014 with 2 generators> ]
gap> time;
30
\endexample

Finally we consider an order which factorises in seven primes and
contains a moderately large prime power. Note that there are 943
non-nilpotent groups of order $288 = 2^5 \cdot 3^2$ while there
are only 90 such groups without normal Sylow subgroup.

\beginexample
gap> flags := rec( nonnilpot := true );;
gap> FrattiniExtensionMethod( 2^5 * 3^2, flags, true );;
gap> time;
656630

gap> flags := rec( nonpnorm := [2,3] );;
gap> FrattiniExtensionMethod( 2^5 * 3^2, flags, true );;
gap> time;
58180
\endexample

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Cephes Mathematical Library

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