Clustering is an unsupervised learning task that aims to partition data into a set of clusters. In many applications, these clusters correspond to real-world constructs (e.g., electoral districts) whose benefit can only be attained by groups when they reach a minimum level of representation (e.g., 50\% to elect their desired candidate). This paper considers the problem of performing k-means clustering while ensuring groups (e.g., demographic groups) have that minimum level of representation in a specified number of clusters. We formulate the problem through a mixed-integer optimization framework and present an alternating minimization algorithm, called MiniReL, that directly incorporates the fairness constraints. While incorporating the fairness criteria leads to an NP-Hard assignment problem within the algorithm, we provide computational approaches that make the algorithm practical even for large datasets. Numerical results show that the approach is able to create fairer clusters with practically no increase in the clustering cost across standard benchmark datasets.