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This article proposes a novel deep clustering model based on the variational autoencoder (VAE), named GamMM-VAE, which can learn latent representations of training data for clustering in an unsupervised manner. Most existing VAE-based deep clustering methods use the Gaussian mixture model (GMM) as a prior on the latent space. We employ a more flexible asymmetric Gamma mixture model to achieve higher quality embeddings of the data latent space. Second, since the Gamma is defined for strictly positive variables, in order to exploit the reparameterization trick of VAE, we propose a transformation method from Gaussian distribution to Gamma distribution. This method can also be considered a Gamma distribution reparameterization trick, allows gradients to be backpropagated through the sampling process in the VAE. Finally, we derive the evidence lower bound (ELBO) based on the Gamma mixture model in an effective way for the stochastic gradient variational Bayesian (SGVB) estimator to optimize the proposed model. ELBO, a variational inference objective, ensures the maximization of the approximation of the posterior distribution, while SGVB is a method used to perform efficient inference and learning in VAEs. We validate the effectiveness of our model through quantitative comparisons with other state-of-the-art deep clustering models on six benchmark datasets. Moreover, due to the generative nature of VAEs, the proposed model can generate highly realistic samples of specific classes without supervised information.