 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/10/c/solution

χ (U^{j}) = 1 + 2 cos (jπ /2) .

Each

U^{j} V

has a planar reflection block of trace zero and third diagonal entry

- 1

, giving character

- 1

. In the chosen element order,

(χ (g_{0}), \dots, χ (g_{7})) = (3, 1, - 1, 1, - 1, - 1, - 1, - 1) .

Equivalently its values on the conjugacy classes

{I}

{U^{2}}

{U, U^{3}}

{V, U^{2} V}

{U V, U^{3} V}

are

3, - 1, 1, - 1, - 1

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