.
For solving such problems, there is beautiful Internet site
https://www.dcode.fr
which has a section devoted to permutations https://www.dcode.fr/permutations-generator.
It asked me what items I want to permute, and I pointed 5 letters A, G, A, I and N.
Then it asked me, if I want to have only DISTINGUISHABLE words as the output, and I answered "Yes".
In response, it generated 60 = words placing them in column.
This output can be ordered in ascending or descending order - I selected "descending", exactly as in a dictionary
(it is also called lexicographic order).
From the list, I copied the 50-th line.
It is the word "NAAIG" , which is your ANSWER.
Probably, I could solve the problem without the help from this solver,
but in this case I decided to save my brain and my time.