# Symmetry group of equilateral triangle

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.

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.
• $G\cong H$ means that $G$ and $H$ are isomorphic as groups. So we may consider them as the same group (since you deal with irreducible representations of groups I assume that you are familiar with isomorphisms. It also seems that you never have accepted any answer:) ). – Dietrich Burde Nov 21 '18 at 10:01
• "It is a non-abelian group and hence isomorphic to $S_3$, since...groups of order 6." Would you be able to elaborate on this sentence? What property of $C_6$ implies that $D_3$ is isomorphic to $S_3$ and why? I am interested in these lower symmetries are connected to $C_6$. Thank you – Blaisem May 25 at 10:01
• @Blaisem There are only two different groups of order $6$, namely $C_6$ and $D_3=S_3$, see here and related links. The definition of $D_3$ can be given with $r=(123)$ and $s=(12)$, so that $D_3=S_3$. – Dietrich Burde May 25 at 11:51