Primitive Groups

A permutation group $G$ acting on $X$ is primitive if $G$ is transitive, and also $G$ does not stabilise any non-trivial partition of $X$ (such partitions are also known as block structures).

Primitive groups are a fundamental building block of many algorithms, as many algorithms use the block-structures of primitive groups to subdivide problems.

Two important primitive groups on a set $X$ are the Natural Symmetric Group and Natural Alternating Group, both acting on $|X|$ points. While these are primitive groups they are often omitted in experiments, and special cased by algorithms which run on primitive groups.

A GAP file containing the generating function is also available.

PrimitiveGenerator := function()
    return AllPrimitiveGroups(NrMovedPoints, [2..4095]);
end;