Federated clustering, an essential extension of centralized clustering for federated scenarios, enables multiple data-holding clients to collaboratively group data while keeping their data locally. In centralized scenarios, clustering driven by representation learning has made significant advancements in handling high-dimensional complex data. However, the combination of federated clustering and representation learning remains underexplored. To bridge this, we first tailor a cluster-contrastive model for learning clustering-friendly representations. Then, we harness this model as the foundation for proposing a new federated clustering method, named cluster-contrastive federated clustering (CCFC). Benefiting from representation learning, the clustering performance of CCFC even double those of the best baseline methods in some cases. Compared to the most related baseline, the benefit results in substantial NMI score improvements of up to 0.4155 on the most conspicuous case. Moreover, CCFC also shows superior performance in handling device failures from a practical viewpoint.