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Then you would say "I and a group of dads". "I and other dads" directly states that you are part of a group (dads).

No it doesn't. It can. "I and a group of dads" creates disassociation where there may be more association to the group than their status as dads.

> No it doesn't.

What? Yes it does. To any native English speaker, “I and other X” means you’re part of X, always.