The SearchPerfectGroups(spec) command searches Maple's database of perfect groups for groups satisfying properties specified in a sequence spec of search parameters. The valid search parameters may be grouped into several classes according to the type of property to which they correspond. These are described in the subsections below.

Use the form = X option to control the form of the output from this command. By default, an expression sequence of IDs for the PerfectGroups 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.

The output = X option controls the type of output produced by the SearchPerfectGroups command. The default output = "list" option causes SearchPerfectGroups to return an expression sequence of outputs. Alternatively, use the output = "iterator" option have SearchPerfectGroups instead return an iterator object that you can use to iterate over the query results one at a time. This is mainly useful in conjunction with the form = "permgroup" option or the form = "fpgroup" option. Finally, the output = "count" option causes SearchPerfectGroups 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 PerfectGroups database.  These may differ from the IDs for the same group if it happens to be present in another database, such as the SmallGroups database, which has its own set of group IDs.

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

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

$1098$

$\mathrm{SearchPerfectGroups}\left(\mathrm{order}=1..500\,'\mathrm{simple}'=\mathrm{false}\,'\mathrm{quasisimple}'\right)$

$\left[120\,1\right],\left[336\,1\right]$

$\mathrm{SearchPerfectGroups}\left(\mathrm{order}=1..500\,'\mathrm{simple}'=\mathrm{false}\,'\mathrm{quasisimple}'\,'\mathrm{form}'=''fpgroup''\right)$

$⟨\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\mid {\mathrm{a3}}^2\,{\mathrm{a1}}^2{\mathrm{a3}}^{-1}\,{\mathrm{a2}}^3\,{\mathrm{a3}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a3}\mathrm{a2}\,\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}⟩,⟨\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\mid {\mathrm{a3}}^2\,{\mathrm{a1}}^2{\mathrm{a3}}^{-1}\,{\mathrm{a2}}^3\,{\mathrm{a3}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a3}\mathrm{a2}\,\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\,{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}{\mathrm{a3}}^{-1}⟩$

$\mathrm{SearchPerfectGroups}\left(\mathrm{order}=1..500\,'\mathrm{simple}'=\mathrm{false}\,'\mathrm{quasisimple}'\,'\mathrm{form}'=''permgroup''\right)$

$⟨\left(1\,2\,5\,3\right)\left(4\,7\,6\,8\right)\left(9\,13\,11\,14\right)\left(10\,15\,12\,16\right)\left(17\,19\,18\,20\right)\left(21\,24\,23\,22\right)\,\left(1\,4\,2\right)\left(3\,5\,6\right)\left(7\,9\,10\right)\left(8\,11\,12\right)\left(13\,16\,17\right)\left(14\,15\,18\right)\left(19\,21\,22\right)\left(20\,23\,24\right)\,\left(1\,5\right)\left(2\,3\right)\left(4\,6\right)\left(7\,8\right)\left(9\,11\right)\left(10\,12\right)\left(13\,14\right)\left(15\,16\right)\left(17\,18\right)\left(19\,20\right)\left(21\,23\right)\left(22\,24\right)⟩,⟨\left(1\,2\,5\,3\right)\left(4\,7\,6\,8\right)\left(9\,13\,11\,14\right)\left(10\,15\,12\,16\right)\,\left(1\,4\,2\right)\left(3\,5\,6\right)\left(7\,9\,10\right)\left(8\,11\,12\right)\,\left(1\,5\right)\left(2\,3\right)\left(4\,6\right)\left(7\,8\right)\left(9\,11\right)\left(10\,12\right)\left(13\,14\right)\left(15\,16\right)⟩$

$\mathrm{SearchPerfectGroups}\left('\mathrm{simple}'=\mathrm{false}\,'\mathrm{quasisimple}'\,'\mathrm{output}'=''count''\right)$

$54$

$\mathrm{it}≔\mathrm{SearchPerfectGroups}\left('\mathrm{simple}'=\mathrm{false}\,'\mathrm{quasisimple}'\,'\mathrm{output}'=''iterator''\,'\mathrm{form}'=''permgroup''\right)$

$\mathrm{it}≔\mathrm{\⟨Iterator\; for\; PerfectGroups\; Query:\; "simple\; =\; false,\; quasisimple",\; with\; 54\; results\⟩}$

$\mathrm{nops}\left(\left[\mathrm{seq}\right]\left(\mathrm{it}\right)\right)$

$54$

$$\begin{gathered} \mathbf{for}\mathrm{id},G\mathbf{in}\mathrm{it}\mathbf{do} \\ t≔\mathrm{GroupOrder}\left(\mathrm{Center}\left(G\right)\right); \\ \mathbf{if}\mathrm{type}\left(t\,'\mathrm{odd}'\right)\mathbf{then} \\ \mathrm{print}\left(\mathrm{id}\right); \\ \mathbf{break} \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$1080,1$

$\mathrm{SearchPerfectGroups}\left('\mathrm{quasisimple}'\,'\mathrm{center}'=4\right)$

$\left[80640\,2\right],\left[80640\,3\right],\left[80640\,4\right],\left[116480\,1\right]$

$\mathrm{SearchPerfectGroups}\left('\mathrm{quasisimple}'\,'\mathrm{center}'=\left[4\,2\right]\right)$

$\left[80640\,2\right],\left[116480\,1\right]$

$\mathrm{IdentifySmallGroup}\left(\mathrm{Center}\left(\mathrm{PerfectGroup}\left(80640\,2\right)\right)\right)$

$4,2$

$\mathrm{IdentifySmallGroup}\left(\mathrm{Alt}\left(5\right)\right)$

$60,5$

$\mathrm{SearchPerfectGroups}\left(\mathrm{socle}=\left[60\,5\right]\right)$

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

$\mathrm{IdentifySmallGroup}\left(\mathrm{PerfectGroup}\left(60\,1\right)\right)$

$60,5$

The GroupTheory[SearchPerfectGroups] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.

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

The output option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.