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We present a theoretical study and experimental realization of a system thatis simultaneously a four-dimensional (4D) Chern insulator and a higher-ordertopological insulator (HOTI). The system sustains the coexistence of(4-1)-dimensional chiral topological hypersurface modes (THMs) and(4-2)-dimensional chiral topological surface modes (TSMs). Our study revealsthat the THMs are protected by second Chern numbers, and the TSMs are protectedby a topological invariant composed of two first Chern numbers, each belonginga Chern insulator existing in sub-dimensions. With the synthetic coordinatesfixed, the THMs and TSMs respectively manifest as topological edge modes (TEMs)and topological corner modes (TCMs) in the real space, which are experimentallyobserved in a 2D acoustic lattice. These TCMs are not related to quantizedpolarizations, making them fundamentally distinctive from existing examples. Wefurther show that our 4D topological system offers an effective way for themanipulation of the frequency, location, and the number of the TCMs, which ishighly desirable for applications.

Four-Dimensional Higher-Order Chern Insulator and Its Acoustic Realization