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## Dih(24) and Sym(4) are not isomorphic

Prove that $D_{24}$ and $S_4$ are not isomorphic.

Solution: We know from Exercise 1.2.3 that $D_{24}$ has 13 elements of order two; namely $r^6$ and $sr^k$ for $0 \leq k < 12$. However, $S_4$ has 9 elements of order two; namely the 2-cycles and products of two 2-cycles.