This question is somewhat ambiguous and really can't be answered. There is no place in Utah called "Great Lake City". The major lake in Utah is called the "Great Salt Lake" and the capitol of Utah is called "Salt Lake City".
It would all depend on the city in California that you are driving from. I live in Utah , and can tell you that from Huntington Beach,CA to Salt Lake City, Ut it is 10 hours driving straight there. 4 hours to Las Vegas and then 6 hours from Las Vegas to Salt Lake City.