1. **Problem Statement:** We need to multiply the symmetry operations \(\sigma_v\) at Cu, Br(1), F(3), F(4) with \(S_3\) at \(C_3\), and then multiply \(\sigma_v\) at Cu, Cl, F(3), F(4) with \(\sigma_h\) at 2Br, Cu, and Cl. We must also determine if these multiplications commute or not. 2. **Relevant Concepts:** - \(\sigma_v\) is a vertical mirror plane symmetry operation. - \(S_3\) is an improper rotation (rotation followed by reflection) of order 3. - \(\sigma_h\) is a horizontal mirror plane. - Multiplication of symmetry operations means performing one operation followed by the other. - Two operations commute if \(AB = BA\). 3. **Step 1: Multiply \(\sigma_v (Cu, Br(1), F(3), F(4)) \times S_3 (C_3)\)** - \(S_3\) combines a rotation about the \(C_3\) axis and reflection through a plane perpendicular to it. - Applying \(S_3\) changes positions of atoms, then \(\sigma_v\) reflects them. - Because \(S_3\) involves rotation and reflection, and \(\sigma_v\) is a vertical reflection, their order affects the result. 4. **Step 2: Multiply \(\sigma_v (Cu, Cl, F(3), F(4)) \times \sigma_h (2Br, Cu, Cl)\)** - \(\sigma_v\) is vertical reflection; \(\sigma_h\) is horizontal reflection. - Reflections about perpendicular planes generally do not commute. - The product of two reflections is a rotation or improper rotation depending on the planes. 5. **Step 3: Check commutation** - For the first multiplication, \(\sigma_v \times S_3 \neq S_3 \times \sigma_v\) because \(S_3\) includes rotation and reflection altering the order. - For the second multiplication, \(\sigma_v \times \sigma_h \neq \sigma_h \times \sigma_v\) since reflections about perpendicular planes do not commute. 6. **Summary:** - Both multiplications do NOT commute. - The products correspond to different symmetry operations depending on the order. **Final answer:** \(\sigma_v (Cu, Br(1), F(3), F(4)) \times S_3 (C_3) \neq S_3 (C_3) \times \sigma_v (Cu, Br(1), F(3), F(4))\) (do not commute). \(\sigma_v (Cu, Cl, F(3), F(4)) \times \sigma_h (2Br, Cu, Cl) \neq \sigma_h (2Br, Cu, Cl) \times \sigma_v (Cu, Cl, F(3), F(4))\) (do not commute).