Symmetry Group of Equilateral Triangle/Cayley Table

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Cayley Table of Symmetry Group of Equilateral Triangle

The Cayley table of the symmetry group of the equilateral triangle can be written:

$\begin{array}{c|ccc|ccc} \circ & e & p & q & r & s & t \\ \hline e & e & p & q & r & s & t \\ p & p & q & e & s & t & r \\ q & q & e & p & t & r & s \\ \hline r & r & t & s & e & q & p \\ s & s & r & t & p & e & q \\ t & t & s & r & q & p & e \\ \end{array}$


where:

\(\displaystyle e\) \(:\) \(\displaystyle (A) (B) (C)\) Identity mapping
\(\displaystyle p\) \(:\) \(\displaystyle (ABC)\) Rotation of $120 \degrees$ anticlockwise about center
\(\displaystyle q\) \(:\) \(\displaystyle (ACB)\) Rotation of $120 \degrees$ clockwise about center
\(\displaystyle r\) \(:\) \(\displaystyle (BC)\) Reflection in line $r$
\(\displaystyle s\) \(:\) \(\displaystyle (AC)\) Reflection in line $s$
\(\displaystyle t\) \(:\) \(\displaystyle (AB)\) Reflection in line $t$


SymmetryGroupEqTriangle.png


Sources