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How deep does school admissions look? Do they ask for DD214? and ask what the misconduct was about?
All schools run a complete background check and this would definitely be flagged.
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What do they ask for to prove i was in the army?
Same as above.
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Is it worth mentioning my military experience?
Yes. You aren't going to be able to "hide" your military experience or your GUHC discharge. Transparency is key here - you do not want to appear as though you are trying to mislead them. You sign the school's honor code or a "statement of integrity" at the end of your application.
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I want to mention it but I am afraid "misconduct" would actually have a negative effect.
The cold hard truth is that it is going to be incredibly difficult to get over this hurdle, especially because it involved "misconduct". What was the misconduct? You can PM me if you don't want to share it on here - I can't really advise you too well without knowing the details.
Broadly speaking, I would tackle it head on. You could talk about your mistakes, what you learned, and what you've done to fix yourself since, and then why you want to transition to b-school.