GAP (GrpConst) - Chapter 7: 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.

gap> ConstructAllGroups( 60 );; time;
4080

In 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 \(p\)-groups which, in turn, might better be constructed by \(p\)-group generation methods.

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

Next 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 ConstructAllGroups which is therefore much more efficient in the next example.

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

Finally 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 \(288 = 2^5 \cdot 3^2\) while there are only 90 such groups without normal Sylow subgroup.

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

7 Examples with Runtimes