we will be in beijing on dec.20 stay for 6night can you recommend me travel itinerary on where to go

Personally, 6 nights are too long to only stay for travel. Anyway, here is a good itinerary for your reference:
D1: Tiananmen Square, Forbidden City
D2: Summer Palace and Jingshan Park
D3: Temple of Heaven and Kung Fu show
D4: Art Zone 798
D5: Badaling or Mutianyu Great Wall
D6: Ming Tombs and Sacred Road
D7: Bird Nest, Water Cube and Wangfujing Pedestrian

Apart from the attractions above, local Happy Valley, Hutongs and Courtyard and Beihai Park are also worth well.

My suggestion is that if you don't mind, go to Tianjin for a visit as well, which is about half an hour bullet train's distance.
More tour packages can be found through this link:
https://www.travelchinaguide.com/package/beijing.htm