I believe the matrix and permutation groups are quite old code, so 
UniqueRepresentation may not have been available at that time. The 
permutation group code does need updating; in particular, it is still an 
old-style parent:

sage: S.__class__.__mro__
(<class 'sage.groups.perm_gps.permgroup_named.SymmetricGroup_with_category'
>,
 <class 'sage.groups.perm_gps.permgroup_named.SymmetricGroup'>,
 <class 'sage.groups.perm_gps.permgroup_named.PermutationGroup_symalt'>,
 <class 'sage.groups.perm_gps.permgroup_named.PermutationGroup_unique'>,
 <class 'sage.structure.unique_representation.CachedRepresentation'>,
 <class 'sage.groups.perm_gps.permgroup.PermutationGroup_generic'>,
 <type 'sage.groups.old.FiniteGroup'>,
 <type 'sage.groups.old.Group'>,
 <type 'sage.structure.parent_gens.ParentWithGens'>,
 <type 'sage.structure.parent_base.ParentWithBase'>,
 <type 'sage.structure.parent_old.Parent'>,
 <type 'sage.structure.parent.Parent'>,
...

(Side note: We also should not be importing things like 
PermutationGroup_generic and PermutationGroupMorphism_id into the global 
namespace.)

I don't see a reason off-hand why these should not be UniqueRepresentation 
subclasses.

Best,
Travis

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