# Category:Dihedral Group D3

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This category contains examples of Dihedral Group D3.

The dihedral group $D_3$ is the symmetry group of the equilateral triangle:

Let $\triangle ABC$ be an equilateral triangle.

We define in cycle notation the following symmetry mappings on $\triangle ABC$:

\(\displaystyle e\) | \(:\) | \(\displaystyle \tuple A \tuple B \tuple C\) | Identity mapping | ||||||||||

\(\displaystyle p\) | \(:\) | \(\displaystyle \tuple {ABC}\) | Rotation of $120 \degrees$ anticlockwise about center | ||||||||||

\(\displaystyle q\) | \(:\) | \(\displaystyle \tuple {ACB}\) | Rotation of $120 \degrees$ clockwise about center | ||||||||||

\(\displaystyle r\) | \(:\) | \(\displaystyle \tuple {BC}\) | Reflection in line $r$ | ||||||||||

\(\displaystyle s\) | \(:\) | \(\displaystyle \tuple {AC}\) | Reflection in line $s$ | ||||||||||

\(\displaystyle t\) | \(:\) | \(\displaystyle \tuple {AB}\) | Reflection in line $t$ |

Note that $r, s, t$ can equally well be considered as a rotation of $180 \degrees$ (in $3$ dimensions) about the axes $r, s, t$.

Then these six operations form a group.

This group is known as the **symmetry group of the equilateral triangle**.

## Pages in category "Dihedral Group D3"

The following 8 pages are in this category, out of 8 total.

### C

- Coset Space forms Partition/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b
- Coset/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b
- Coset/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b/Left Cosets
- Coset/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b/Right Cosets