US Cash Products-Group Three gasoline falls sharply, demand weak

* Thin trade contributes to sharp Group Three gasoline swing * Gulf Coast gasoline slides in thin trade * Harbor distillates move lower HOUSTON, Dec 27 (Reuters) - Midwest Group Three gasoline differentials plunged by 6.00 cents per gallon in a thinly traded market on Thursday on weak demand, tracing declines in Gulf Coast gasoline, traders said. Group Three gasoline fell to 36.00/34.00 cents under the January RBOB gasoline futures contract on the New York Mercantile Exchange. F