Assuming you lost or did not have the DMV hearing, and it sounds like that is the case, then the DMV suspension begins on the arrest date and you should be eligible for a full license, depending on the conditions of the reckless.
If you did not have DMV hearing, then usually yes.
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I believe that the DUI citation acts as your temporary driver's license for ten days after the arrest, and that the administrative suspension would begin after that tenth day.
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