 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/8/c/solution

D_{6} = ⟨ r, s ∣ r^{6} = s^{2} = 1, srs = r^{- 1} ⟩

, where

r

rotates through

π /3

. There are six rotations and six reflections, so the order is 12. Conjugation by

s

reverses a rotation, and conjugation by

r

sends

r^{j} s

r^{j + 2} s

. Consequently the conjugacy classes and their sizes are

{1} (1), {r^{3}} (1), {r, r^{5}} (2), {r^{2}, r^{4}} (2), {s, r^{2} s, r^{4} s} (3), {rs, r^{3} s, r^{5} s} (3) .

The last two classes distinguish the reflection axes through opposite vertices and through opposite edge midpoints. Their sizes sum to 12.

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