Count the total number of groups in the database.
Find all the nilpotent groups of order .
Find all the non-Abelian groups of order , as permutation groups.
Find those finite groups in the database whose Sylow -subgroup is the cyclic group of order , and whose Sylow -subgroup is the small group 27/4.
Find the perfect groups of order at most .
Verify that there is no non-Abelian simple group with order less than . The NULL expression sequence as output indicates there are no groups in the database satisfying the indicated combination of properties.
Another way to verify that there are no non-abelian simple groups with order less than is to use the output = "count" option.
Find all the non-Abelian simple groups in the database.
Find the groups up to order whose Sylow -subgroup has order .
Find the non-nilpotent groups of order with perfect order classes.
Find the perfect groups in the database that are not simple.
Find the non-Abelian groups of order .
Show that neither of these occurs as the Frattini subgroup of any group in the database. (In fact, neither occurs as the Frattini subgroup of any finite group.)
Construct an iterator for the non-abelian groups of order ; the form = "permgroup" option instructs SearchSmallGroups to construct an iterator yielding permutation groups rather than the default IDs.
Use the constructed iterator to survey the nilpotency classes of these groups.
Find the groups of order at most whose Sylow -subgroup and Sylow -subgroup is normal. The nsylow[p] option describes the number of non-trivial Sylow -subgroups and a Sylow -subgroup is normal if there is only one Sylow -subgroup.
Find the perfect, but non-simple groups in the database whose central quotient has order 168.
Two ways to isolate the non-Abelian simple groups in the database are shown as follows
The form = "count" option is particularly handy for developing visualizations depicting the relative frequency of groups satisfying particular properties.