The SearchSmallGroups(spec) command searches Maple's small groups database for groups satisfying properties specified in a sequence spec of search parameters. The valid search parameters may be grouped into several classes, as follows.

Use the form = X option to control the form of the output from this command. By default, an expression sequence of IDs for the Small groups database is returned. This is the same as specifying form = "id". To have an expression sequence of groups, either permutation groups, or finitely presented groups, use either the form = "permgroup" or form = "fpgroup" options, respectively. Finally, the form = "count" option causes SearchSmallGroups to return just the number of groups in the database satisfying the constraints implied by the search parameters.

Note that the IDs returned in the default case are the IDs of the groups within the SmallGroups 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.

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

$\mathrm{SearchSmallGroups}\left('\mathrm{form}'=''count''\right)$

$92804$

$\mathrm{SearchSmallGroups}\left('\mathrm{order}'=24\,'\mathrm{nilpotent}'\right)$

$\left[24\,2\right],\left[24\,9\right],\left[24\,10\right],\left[24\,11\right],\left[24\,15\right]$

$\mathrm{SearchSmallGroups}\left('\mathrm{order}'=10\,'\mathrm{abelian}'=\mathrm{false}\,'\mathrm{form}'=''permgroup''\right)$

$⟨\left(1\,2\right)\left(3\,6\right)\left(4\,5\right)\left(7\,10\right)\left(8\,9\right)\,\left(1\,3\,7\,8\,4\right)\left(2\,5\,9\,10\,6\right)⟩$

$\mathrm{SearchSmallGroups}\left(\mathrm{sylow}\left[2\right]=\left[4\,1\right]\,\mathrm{sylow}\left[3\right]=\left[27\,4\right]\right)$

$\left[108\,9\right],\left[108\,14\right]$

$\mathrm{SearchSmallGroups}\left('\mathrm{order}'=1..100\,'\mathrm{perfect}'\right)$

$\left[1\,1\right],\left[60\,5\right]$

$\mathrm{SearchSmallGroups}\left('\mathrm{simple}'\&comma;'\mathrm{abelian}'=\mathrm{false}\&comma;'\mathrm{order}'<60\right)$

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

$\left[60\&comma;5\right],\left[168\&comma;42\right],\left[360\&comma;118\right],\left[504\&comma;156\right]$

$\mathrm{SearchSmallGroups}\left('\mathrm{order}'=1..24\&comma;'\mathrm{sylow}'\left[2\right]=4\right)$

$\left[4\&comma;1\right],\left[4\&comma;2\right],\left[12\&comma;1\right],\left[12\&comma;2\right],\left[12\&comma;3\right],\left[12\&comma;4\right],\left[12\&comma;5\right],\left[20\&comma;1\right],\left[20\&comma;2\right],\left[20\&comma;3\right],\left[20\&comma;4\right],\left[20\&comma;5\right]$

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

$\left[1\&comma;1\right],\left[120\&comma;5\right],\left[336\&comma;114\right]$

$\mathrm{SearchSmallGroups}\left('\mathrm{order}'=8\&comma;'\mathrm{abelian}'=\mathrm{false}\right)$

$\left[8\&comma;3\right],\left[8\&comma;4\right]$

$\mathrm{SearchSmallGroups}\left('\mathrm{frattinisubgroup}'=\left[8\&comma;3\right]\right)$

$\mathrm{SearchSmallGroups}\left('\mathrm{frattinisubgroup}'=\left[8\&comma;4\right]\right)$

$L≔\left[\mathrm{SearchSmallGroups}\right]\left('\mathrm{maxelementorder}'=5\right)\&colon;$

$\mathrm{map}\left(\mathrm{irem}\&comma;\mathrm{map2}\left(\mathrm{op}\&comma;1\&comma;L\right)\&comma;5\right)$

$\left[0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\&comma;0\right]$

$\mathrm{seq}\left(\mathrm{SearchSmallGroups}\left('\mathrm{maxelementorder}'=i\&comma;\mathrm{form}=''count''\right)\&comma;i=2..10\right)$

$8,18,31413,19,361,21,24625,182,149$

$\mathrm{SearchSmallGroups}\left('\mathrm{order}'=1..32\&comma;'\mathrm{nsylow}'\left[2\right]=1\&comma;'\mathrm{nsylow}'\left[3\right]=1\right)$

$\left[6\&comma;2\right],\left[12\&comma;2\right],\left[12\&comma;5\right],\left[18\&comma;2\right],\left[18\&comma;5\right],\left[24\&comma;2\right],\left[24\&comma;9\right],\left[24\&comma;10\right],\left[24\&comma;11\right],\left[24\&comma;15\right],\left[30\&comma;4\right]$

$\mathrm{SearchSmallGroups}\left('\mathrm{simple}'=\mathrm{false}\&comma;'\mathrm{perfect}'\&comma;'\mathrm{centralquotient}'=168\right)$

$\left[336\&comma;114\right]$

$\mathrm{Statistics}:-\mathrm{PieChart}\left(\left[\mathrm{seq}\right]\left(i=\mathrm{SearchSmallGroups}\left('\mathrm{fittinglength}'=i\&comma;'\mathrm{form}'=''count''\right)\&comma;i=1..3\right)\right)$

$\mathrm{plots}:-\mathrm{display}\left(\left[\mathrm{Statistics}:-\mathrm{PieChart}\left(\left[\mathrm{seq}\right]\left(i=\mathrm{SearchSmallGroups}\left('\mathrm{derivedlength}'=i\&comma;'\mathrm{form}'=''count''\right)\&comma;i=1..5\right)\&comma;'\mathrm{annular}'=1\right)\&comma;\mathrm{Statistics}:-\mathrm{PieChart}\left(\left[\mathrm{seq}\right]\left(i=\mathrm{SearchSmallGroups}\left('\mathrm{nilpotencyclass}'=i\&comma;'\mathrm{form}'=''count''\right)\&comma;i=1..7\right)\&comma;'\mathrm{annular}'=2\right)\&comma;\mathrm{Statistics}:-\mathrm{PieChart}\left(\left[\mathrm{seq}\right]\left(i=\mathrm{SearchSmallGroups}\left('\mathrm{compositionlength}'=i\&comma;'\mathrm{form}'=''count''\right)\&comma;i=1..8\right)\&comma;'\mathrm{annular}'=3\right)\right]\right)$

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

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

The spec parameter was updated in Maple 2019.

The form option was updated in Maple 2019.

The GroupTheory[SearchSmallGroups] command was updated in Maple 2020.