The SearchTransitiveGroups( spec ) command searches Maple's transitive groups database for groups satisfying properties specified in a sequence spec of search parameters. This allows you to locate examples of transitive permutation groups that have specific, supported properties, or combinations of those properties.

The output option controls the type of output produced by the SearchTransitiveGroups command, while the form option controls (depending upon the form selected) what form the content of that output has.

Use the output = X option, for which the default is the string "list", to control what the SearchTransitiveGroups command produces. By default (with output = "list"), a sequence of either identifiers or permutation groups is returned. Specifying output = "iterator" causes SearchTransitiveGroups to instead return an iterator object which produces either identifiers or permutation groups as you iterate over the object. The output = "count" option instructs the SearchTransitiveGroups command to return just the number of groups in the database that satisfy the search query indicated by spec. In this case, the form option has no effect, as the "form" of the output is necessarily just a non-negative integer.

Use the form = X option to control the form of the output from this command when the output is either a sequence or an iterator object. By default, an expression sequence of IDs for the TransitiveGroups database is returned. This is the same as specifying form = "id". To have an expression sequence of permutation groups, use the form = "permgroup"  option.

Note that the IDs returned in the default case are the IDs of the groups within the TransitiveGroups database.  These may differ from the IDs for the same group if it happens to be present in another database, which has its own set of group IDs. In particular, the first member of the TransitiveGroups database ID is the degree of the group, not its order.

So, for example, the symmetric group of degree $3$ appears in the database of transitive groups with ID equal to (3, 2), but appears also in the database of small groups with ID equal to (6, 1).

The valid search parameters may be grouped into several classes, as follows.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{SearchTransitiveGroups}\left('\mathrm{output}'=''count''\right)$

$162594$

$\mathrm{id}≔\mathrm{SearchTransitiveGroups}\left('\mathrm{degree}'=6\,'\mathrm{transitivity}'=2\right)$

$\mathrm{id}≔\left[6\,12\right]$

$G≔\mathrm{TransitiveGroup}\left(\mathrm{op}\left(\mathrm{id}\right)\right)$

$G≔⟨\left(1\,2\,3\,4\,6\right)\,\left(1\,4\right)\left(5\,6\right)⟩$

$\mathrm{Degree}\left(G\right)$

$6$

$\mathrm{id}≔\mathrm{SearchTransitiveGroups}\left('\mathrm{order}'=24\&comma;1<'\mathrm{transitivity}'\right)$

$\mathrm{id}≔\left[4\&comma;5\right]$

$\mathrm{Transitivity}\left(\mathrm{TransitiveGroup}\left(\mathrm{op}\left(\mathrm{id}\right)\right)\right)$

$4$

$\mathrm{SearchTransitiveGroups}\left('\mathrm{degree}'=6\&comma;'\mathrm{regular}'\&comma;'\mathrm{form}'=''permgroup''\right)$

$⟨\left(1\&comma;6\&comma;5\&comma;4\&comma;3\&comma;2\right)⟩,⟨\left(1\&comma;3\&comma;5\right)\left(2\&comma;4\&comma;6\right)\&comma;\left(1\&comma;4\right)\left(2\&comma;3\right)\left(5\&comma;6\right)⟩$

$\mathrm{SearchTransitiveGroups}\left('\mathrm{quasisimple}'\&comma;'\mathrm{simple}'=\mathrm{false}\right)$

$\left[16\&comma;715\right],\left[18\&comma;262\right],\left[24\&comma;201\right],\left[24\&comma;2947\right],\left[24\&comma;18440\right]$

$\mathrm{SearchTransitiveGroups}\left('\mathrm{primitive}'\&comma;'\mathrm{output}'=''count''\right)$

$294$

$\mathrm{SearchTransitiveGroups}\left('\mathrm{degree}'=24\&comma;'\mathrm{output}'=''count''\right)=\mathrm{NumTransitiveGroups}\left(24\right)$

$25000=25000$

$\mathrm{SearchTransitiveGroups}\left('\mathrm{semiprimitive}'\&comma;'\mathrm{fittingsubgroup}'=8\right)$

$\left[8\&comma;12\right],\left[8\&comma;23\right],\left[8\&comma;25\right],\left[8\&comma;36\right],\left[8\&comma;48\right],\left[24\&comma;7\right],\left[24\&comma;9\right],\left[24\&comma;22\right]$

$\mathrm{SearchTransitiveGroups}\left('\mathrm{centre}'=\left[8\&comma;1\right]\right)$

$\left[8\&comma;1\right],\left[16\&comma;16\right],\left[16\&comma;22\right],\left[16\&comma;114\right],\left[16\&comma;124\right],\left[16\&comma;289\right],\left[24\&comma;32\right],\left[24\&comma;85\right],\left[24\&comma;322\right],\left[24\&comma;936\right],\left[24\&comma;1992\right],\left[24\&comma;4206\right],\left[24\&comma;7020\right]$

The GroupTheory[SearchTransitiveGroups] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.

The GroupTheory[SearchTransitiveGroups] command was updated in Maple 2019.