Iterate through the finite groups in MAGMA

Question

How may I iterate over the finite groups in MAGMA of orders between $1$ and $N$ (where possible), and compute a given property of them?

Answer 1

Magma has a database of small groups, see http://magma.maths.usyd.edu.au/magma/handbook/text/777:

Small Groups: This database is constructed by Hans Ulrich Besche, Heiko Dietrich, Bettina Eick, Eamonn O'Brien, and Eileen Pan [BE99a], [BEO01], [BE99b], [O'B90], [BE01], [O'B91], [MNVL04], [OVL05], [DE05], [DEP22], and contains the following groups:

• All groups of order up to 2000, excluding the groups of order 1024.
• The groups whose order is the product of at most 4 primes.
• The groups of order dividing p^7 for p a prime.
• The groups of order 3^8.
• The groups of order q^n p, where q^n is a prime-power dividing 2^8, 3^6, 5^5 or 7^4 and p is a prime different to q.
• The groups of square-free order.

You should be able to use the function SmallGroups(S: parameters) : [RngIntElt] -> [* Grp *] for your purposes, e.g.

for grp in SmallGroups([ 1 .. N ]) do
    // compute some property
end for;

If you are only after the groups satisfying a given property, there is a nice example of that about half-way down this page in the Magma handbook http://magma.maths.usyd.edu.au/magma/handbook/text/779.