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Existing group recommendation models often have a monotonous approach when solving user representation, and only simple social relationships between users are utilized. This makes user representation inaccurate and most models do not consider the impact of user common intention and social interaction on group preferences. As a result, recommended items are not aligned with user needs. To address these issues, a new group recommendation model based on user common intention and social interaction (GR-UCISI) is proposed. Firstly, a user intention separation model that combines user-item interaction history with social interaction is constructed. Graph neural networks are utilized to collect user-item interaction and social interaction information, and to solve user intention and item representation. Secondly, by utilizing the social network random walk algorithm and the [K-means] clustering algorithm, users can be grouped. User group, user intention and group intention aggregation process are combined to obtain group common intention representation. Finally, group common intention representation and item representation are calculated to obtain the list of recommended items for the group. This method fully considers the impact of user individuality and commonality among group members on group preferences. It also utilizes social relationships to alleviate the problem of data sparsity and improve model performance. The experimental results show that compared with the model with the best recommendation effect of nine models, on the Gowalla dataset, the Precision and NDCG of the GR-UCISI model are increased by 3.01% and 5.26% respectively, on the Yelp-2018 dataset, the Precision and NDCG of the GR-UCISI model are increased by 2.96% and 1.12% respectively.