Imagine that I have a two-qubit system, and one of the qubits performs single-qubit rotations from a subgroup of SU(2), specifically 2O. In this subgroup, the single-qubit rotations are represented by H, S,X, Y, Z, and I, where I denotes no rotation. Additionally, I have a CNOT gate for this system. I am wondering how to show numerically that this subgroup and the CNOT gate together form a finite group.
P.S. I understand that this may seem like a trivial question, but I am also trying to grasp the methodology for solving these types of problems, particularly in studying groups and determining whether they are infinite or finite. If someone can provide a comprehensive answer to this question, it will help me understand the approach and process for handling more complex groups as well. I kindly request that any response refrain from phrases such as "with H and T gate, they perform a universal gate set, so if you include them this will perform infinite group" or "it is obvious this will not give us a universal gate set without the T gate and H, so it is finite.
