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Alternatively, you can build your own using the rapidly-evolving Search Kit.
For example, try this on dmaster.demo.civicrm.org
(screenshot added for posterity)
Results:
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If you take one more step, then the solution is easier. When you click on Contacts as you describe you should see an Advanced Search link to the right
OR you could simply initiate your search via Adv Search and choose the required Group via the Groups field.
Then, in order to see different columns, you need to create a Profile that has the fields in it that ...
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