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 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 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.

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 in seven primes and
contains a moderately large prime power. Note that there are 943
non-nilpotent groups of order `288 = 2 ^{5} cdot3^{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

grpconst manual

January 2020