How to create a morphism between a permutation groups and a matrix groups by explicitley giving the map?

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First, phi(x) does not makes much sense since x benongs to the symbolic ring, and has nothing to do with your group H.

However, i agree that the following seems and issue:

sage: phi(r)
RuntimeError: Gap produced error output
Error, usage: Image(<map>), Image(<map>,<elm>), Image(<map>,<coll>)

   executing __SAGE_LAST__:="__SAGE_LAST__";;Image(\$sage11,\$sage9);;

And even:

sage: phi(H(r))
RuntimeError: Gap produced error output
Error, usage: Image(<map>), Image(<map>,<elm>), Image(<map>,<coll>)

   executing __SAGE_LAST__:="__SAGE_LAST__";;Image(\$sage21,\$sage20);;

Thanks for reporting, someone should have a deeper look and see what does gap return to understand.