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Magma contains the means to construct all p-groups of order pn
where n≤7. This section describes the functions for accessing these
constructions. The data used in the constructions was supplied by
Hans Ulrich Besche, Bettina Eick, Eamonn O'Brien, Mike Newman
and Michael Vaughan-Lee
[BE99a], [BEO01], [BE99b], [O'B90], [BE01], [O'B91], [MNVL04], [OVL05].
Produce a sequence of groups of order pn satisfying the conditions
specified by the following parameters. The restrictions on the order
are n ≤7 or p=2 and n≤9.
Rank: SetEnum Default: {1, ... n}
All groups returned will have Frattini quotient rank in Rank.
This parameter may also be set to a single integer.
Class: SetEnum Default: {1, ... n}
All groups returned will have p-class in Class.
This parameter may also be set to a single integer.
Select: Program Default: true
The parameter must be set to a program returning either true or false
when given a p-group satisfying the above conditions.
All groups G returned will then satisfy Select(G) eq true.
Limit: RngIntElt Default: 0
If Limit is set to a positive number n, then
the program may end its search and return when there are at least n groups
found.
Count the number of groups of order pn satisfying the conditions
specified by the parameters. The parameters are the same as for
SearchPGroups, except that the Limit parameter is ignored.
We search the groups of order 197 for specific examples.
There are, in total, 9380741 groups with this order.
We start with a search for those of rank 5, class 3, and exponent 19.
Since we do not set the Limit parameter, we will get a sequence
containing all the examples.
> time Q := SearchPGroups(19, 7:Rank := 5, Class := 3,
> Select := func<G|IsPrime(Exponent(G))> );
Time: 0.050
> #Q;
4
> Q[1];
GrpPC of order 893871739 = 19^7
PC-Relations:
$.2^$.1 = $.2 * $.6,
$.6^$.1 = $.6 * $.7
This time we limit the number returned.
> time Q := SearchPGroups(19, 7:Rank := 4, Class := {3,4},
> Select := func<G|IsPrime(Exponent(G))>, Limit := 5);
Time: 13.090
> #Q;
5
> [pClass(G):G in Q];
[ 3, 3, 3, 3, 3 ]
> time Q4 := SearchPGroups(19, 7:Rank := 4, Class := 4,
> Select := func<G|IsPrime(Exponent(G))>, Limit := 5);
Time: 0.150
> #Q4;
6
Note that the limit is not always adhered to exactly.
We can also count the number of groups with our property.
> time CountPGroups(19, 7:Rank := 4, Class := {3,4},
> Select := func<G|IsPrime(Exponent(G))>);
Time: 334.720
43
> time CountPGroups(19, 7:Rank := 4, Class := 4,
> Select := func<G|IsPrime(Exponent(G))>);
10
Time: 0.310
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