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. These are described in the sections below.

The output option controls the type of output produced by the SearchSmallGroups 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 SearchSmallGroups command produces. By default (with output = "list"), a sequence of either identifiers or groups is returned. Specifying output = "iterator" causes SearchSmallGroups to instead return an iterator object which produces either identifiers or groups as you iterate over the object. The output = "count" option instructs the SearchSmallGroups 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. By default, an expression sequence or iterator of IDs for the small groups database is returned. This is the same as specifying form = "id". To have an expression sequence of or iterator over groups, either permutation groups, or finitely presented groups, use either the form = "permgroup" or form = "fpgroup" options, respectively.

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.

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

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

$92804$

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

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

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

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

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

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

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

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

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

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

$0$

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

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

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

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

$\mathrm{SearchSmallGroups}\left('\mathrm{order}'=24\&comma;'\mathrm{perfectorderclasses}'\&comma;'\mathrm{nilpotent}'=\mathrm{false}\right)$

$\left[24\&comma;1\right],\left[24\&comma;3\right],\left[24\&comma;7\right]$

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

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

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

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

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

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

$\mathrm{it}≔\mathrm{SearchSmallGroups}\left('\mathrm{order}'=32\&comma;'\mathrm{abelian}'=\mathrm{false}\&comma;'\mathrm{form}'=''permgroup''\&comma;'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\&lang;Iterator\; for\; SmallGroups\; Query:\; "order\; =\; 32,\; abelian\; =\; false",\; with\; 44\; results\&rang;}$

$\mathrm{seq}\left(\mathrm{NilpotencyClass}\left(G\right)\&comma;G=\mathrm{it}\right)$

$2,2,2,3,3,3,3,3,3,2,3,3,3,2,4,4,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,2,2,2,2,2$

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

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

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

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

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

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

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

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

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

$\mathrm{evalb}\left(\mathrm{SearchSmallGroups}\left('\mathrm{simple}'\&comma;'\mathrm{abelian}'=\mathrm{false}\right)=\mathrm{SearchSmallGroups}\left('\mathrm{simple}'\&comma;'\mathrm{generationrank}'=2\right)\right)$

$\mathrm{true}$

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

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

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.