If it is Tableau Desktop, I am not aware of that option (though someone may come up with cool idea, lets see). However, you may set up a Welcome Dashboard where you can list all departments and apply action filter to direct them to appropriate dashboard. Will that help you?
I'll look into that. That sounds like a definite solution if there's no actual "default by group" option available.
Absolutely. Also, in case you need further help with action filter, you may want to check this link that gives exactly how you can achieve my approach. Creating a Table of Contents to Navigate to Other Dashboards | Tableau Software. Hope it will help to guide to right path. Thanks