This work aims to address the group activity recognition problem by exploring human motion characteristics. Traditional methods hold that the motions of all persons contribute equally to the group activity, which suppresses the contributions of some relevant motions to the whole activity while overstating some irrelevant motions. To address this problem, we present a Spatio-Temporal Context Coherence (STCC) constraint and a Global Context Coherence (GCC) constraint to capture the relevant motions and quantify their contributions to the group activity, respectively. Based on this, we propose a novel Coherence Constrained Graph LSTM (CCG-LSTM) with STCC and GCC to effectively recognize group activity, by modeling the relevant motions of individuals while suppressing the irrelevant motions. Specifically, to capture the relevant motions, we build the CCG-LSTM with a temporal confidence gate and a spatial confidence gate to control the memory state updating in terms of the temporally previous state and the spatially neighboring states, respectively. In addition, an attention mechanism is employed to quantify the contribution of a certain motion by measuring the consistency between itself and the whole activity at each time step. Finally, we conduct experiments on two widely-used datasets to illustrate the effectiveness of the proposed CCG-LSTM compared with the state-of-the-art methods.