A community structure is a fundamental property of complex networks and its detection plays an important role in exploring and understanding such networks. Due to its great interpretability, a symmetric and non-negative matrix factorization (SNMF) model is frequently adopted to perform community detection tasks. However, it adopts a single latent factor (LF) matrix to construct the approximation of a given undirected matrix to ensure its absolute symmetry at the expense of shrinking its solution space. This paper proposes a symmetry-constrained NMF (SCNMF) method, with two-fold ideas: a) modeling the approximate symmetry of an undirected network by introducing an equality-constraint on LF matrices into an NMF framework; and b) using graph-regularization to extract the features regarding the intrinsic geometric structure of a network. Extensively empirical studies on six real-world social networks from industrial applications demonstrate that the proposed SCNMF-based detector achieves higher accuracy for community detection than state-of-the-art models.