Symmetry group of equilateral triangle

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I have read at some places that the symmetry of equilateral triangle is C3v as well as some places mention it to be D3.

The group tables for these two groups differ, hence they are not isomorphic.

Yet both these groups define symmetry of same shape.

Please, explain what is going on.

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1 Answer

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The symmetry group of an equilateral triangle is the dihedral group $D_3$ with $6$ elements. It is a non-abelian group and hence isomorphic to $S_3$, since $C_6$ is abelian and there are only two different groups of order $6$. So there is one and only one symmetry group of the regular $3$-gon up to isomorphism. In particular, $C_{3v}\cong D_3$.

Reference: see page $105$ here.

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