The perfect groups library contains all the perfect groups of order less than or equal to 1000000, except the groups of the orders 61440, 86016, 122880, 172032, 245760, 344064, 368640, 491520, 688128, 737280 and 983040. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders, a complete and irredundant list of isomorphism type representatives of groups is given. These groups are available as permutation groups and as finitely presented groups.

The PerfectGroup( n, d ) command returns the $d$-th perfect group of order $n$ in the perfect groups library.  The value of $n$ must be at most 1000000 and not among those exceptions listed above.  The value of $d$ must be less than or equal to the number of perfect groups of order $n$. You can also use the syntax PerfectGroup( [ n, d ] ), which is equivalent to PerfectGroup( n, d ).

By default, PerfectGroup( n, d ) returns a permutation group. By passing the option form = "fpgroup", Maple will instead return the $d$-th perfect group of order $n$ as a finitely presented group.

The NumPerfectGroups( n ) command returns the number of perfect groups of order $n$, where $n$ is a valid order.  If $n$ is less than or equal to 1000000 and not among the listed exceptions, then $0$ is returned; otherwise, an exception is raised.

The AllPerfectGroups( n ) command by default returns a list of all the perfect groups of order $n$, where $n$ is a valid order, or a range of orders. You can request that an iterator be returned by specifying the option output = "iterator". The form = "fpgroup" or form = "permgroup" option is also available for the AllPerfectGroups command with the same meaning as for the PerfectGroup command.

Use the SearchPerfectGroups command to search for perfect groups in the database satisfying combinations of supported properties which have been pre-computed.

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

$\mathrm{NumPerfectGroups}\left(60\right)$

$1$

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

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

$\mathrm{PerfectGroup}\left(\left[60\,1\right]\,\mathrm{form}=''permgroup''\right)$

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

$\mathrm{PerfectGroup}\left(60\,1\,\mathrm{form}=''fpgroup''\right)$

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

$L≔\mathrm{AllPerfectGroups}\left(1920\,'\mathrm{form}'=''permgroup''\right)\:$

$\mathrm{andmap}\left(\mathrm{IsPerfect}\,L\right)$

$\mathrm{true}$

$L≔\mathrm{AllPerfectGroups}\left(60..170\right)$

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

$\mathrm{map}\left(\mathrm{GroupOrder}\,L\right)$

$\left[60\,120\,168\right]$

$\mathrm{it}≔\mathrm{AllPerfectGroups}\left(20160\,'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\⟨Perfect\; Groups\; Iterator:\; 20160/1\; ..\; 20160/5\⟩}$

$$\begin{gathered} \mathbf{for}\mathrm{id},G\mathbf{in}\mathrm{it}\mathbf{do} \\ \mathbf{if}\mathbf{not}\mathrm{IsSimple}\left(G\right)\mathbf{then} \\ \mathrm{print}\left(\left[\mathrm{id}\right]\right) \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$\left[20160\,1\right]$

$\left[20160\,2\right]$

$\left[20160\,3\right]$

Derek F. Holt and W. Plesken, Perfect Groups, Oxford Math. Monographs, Oxford University Press, 1989.

The GroupTheory[PerfectGroup], GroupTheory[NumPerfectGroups] and GroupTheory[AllPerfectGroups] commands were introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.