Hi everyone,

I'm Sudarshan Dandgawal, a student preparing my Google Summer of Code 
(GSoC) proposal for SageMath. I have been exploring Issue #34504, which 
addresses inconsistencies in direct product, semidirect product, and direct 
product permgroups for permutation groups.

Currently, these methods do not properly retain the original domains of the 
input groups when computing their products. This leads to incorrect results 
when working with groups on disjoint domains. My proposed solution involves:

   1. 
   
   Ensuring the resulting permutation group correctly reflects the union of 
   the original domains.
   2. 
   
   Implementing proper relabeling mechanisms to maintain consistency.
   3. 
   
   Adding test cases to verify correctness.
   
I'd love to get feedback from the community on my approach before 
finalizing my GSoC proposal. Specifically:

   - 
   
   Are there edge cases I should consider when modifying these methods?
   - 
   
   Would a specific implementation strategy be preferred for handling 
   domain mappings?
   - 
   
   Any additional insights or potential challenges I should be aware of?
   
I’d greatly appreciate any thoughts, suggestions, or guidance on refining 
my proposal. Looking forward to your feedback!

Best,
Sudarshan Dandgawal

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