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GroupTheory

 PerfectGroup
 retrieve a group from the database of perfect groups
 NumPerfectGroups
 return the number of perfect groups of a given order
 AllPerfectGroups
 retrieve all perfect groups of a given order

 Calling Sequence PerfectGroup( n, d ) PerfectGroup( [ n, d ] ) PerfectGroup( n, d, f ) PerfectGroup( [ n, d ], f ) NumPerfectGroups( n ) AllPerfectGroups( n ) AllPerfectGroups( n, f )

Parameters

 n - a positive integer d - a positive integer f - optional equation: form=fpgroup or form=permgroup (the default)

Description

 • The perfect groups library contains all the perfect groups of order at 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 ).
 • 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 returns a list of all the perfect groups of order $n$, where $n$ is a valid order.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{NumPerfectGroups}\left(60\right)$
 ${1}$ (1)
 > $\mathrm{PerfectGroup}\left(60,1\right)$
 $⟨\left({1}{,}{2}\right)\left({4}{,}{5}\right){,}\left({2}{,}{3}{,}{4}\right)⟩$ (2)
 > $\mathrm{PerfectGroup}\left(\left[60,1\right],\mathrm{form}="permgroup"\right)$
 $⟨\left({1}{,}{2}\right)\left({4}{,}{5}\right){,}\left({2}{,}{3}{,}{4}\right)⟩$ (3)
 > $\mathrm{PerfectGroup}\left(60,1,\mathrm{form}="fpgroup"\right)$
 $⟨{}{\mathrm{a1}}{,}{\mathrm{a2}}{}{\mid }{}{{\mathrm{a1}}}^{{2}}{,}{{\mathrm{a2}}}^{{3}}{,}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{a1}}{}{\mathrm{a2}}{}⟩$ (4)
 > $L≔\mathrm{AllPerfectGroups}\left(1920,'\mathrm{form}'="permgroup"\right):$
 > $\mathrm{andmap}\left(\mathrm{IsPerfect},L\right)$
 ${\mathrm{true}}$ (5)

References

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

Compatibility

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