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This paper presents a novel multitask learning framework for palmprint biometrics, which optimizes classification and hashing branches jointly. The classification branch within our framework facilitates the concurrent execution of three distinct tasks: identity recognition and classification of soft biometrics, encompassing gender and chirality. On the other hand, the hashing branch enables the generation of palmprint hash codes, optimizing for minimal storage as templates and efficient matching. The hashing branch derives the complementary information from these tasks by amalgamating knowledge acquired from the classification branch. This approach leads to superior overall performance compared to individual tasks in isolation. To enhance the effectiveness of multitask learning, two additional modules, an attention mechanism module and a customized gate control module, are introduced. These modules are vital in allocating higher weights to crucial channels and facilitating task-specific expert knowledge integration. Furthermore, an automatic weight adjustment module is incorporated to optimize the learning process further. This module fine-tunes the weights assigned to different tasks, improving performance. Integrating the three modules above has shown promising accuracies across various classification tasks and has notably improved authentication accuracy. The extensive experimental results validate the efficacy of our proposed framework.