Summary: In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the $IS$ $(I = 1/2$ and $S = 1/2)$ spin system, two-qubit CNOT gate is represented by a $4 \times 4$ matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the $IS$ $(I = 3/2, S = 3/2)$ spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the $IS$ $(I = 3/2, S = 3/2)$ spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the $IS$ $(I = 3/2, S = 3/2)$ spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.