Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 8D / b / iv / Solution

The action on the four diagonals identifieswhere the $C_2$ factor is generated by central inversion. An element of order three must have trivial $C_2$ component and a three-cycle in $S_4$. Since all three-cycles in $S_4$ are conjugate,Elements of order two are not all conjugate. For example, $-I$ is central and therefore has a singleton conjugacy class, while a reflection is a different element of order two. Thus

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