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That Garmin should power On, and it should charge OK. Check the power adapter that you use to charge it with, or if possible, use a different power source or power adapter, to make sure it is not a fault with the adapter.
(Of course, if the adapter you are trying to charge with came with the Garmin Nuvi 68, then it too is covered by this failure of the unit as a whole, under ... see below...)
After that, you may assume that the warranty applies and it is a failure of warranty-covered equipment from Garmin. So, applying to Garmin for a repair is the right thing to do.
Applying to Garmin for repair (under warranty) is done by going to
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