Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table

Cayley Table for Dihedral Group $D_4$

The Cayley table for the dihedral group $D_4$:

$D_4 = \set {\mathbf I, -\mathbf I, \mathbf A, -\mathbf A, \mathbf B, -\mathbf B, \mathbf C, -\mathbf C}$

under the operation of conventional matrix multiplication, where:

$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

\qquad \mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

can be presented as:

$\begin{array}{r|rrrrrrrr}

          &  \mathbf I &  \mathbf A &  \mathbf B &  \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C \\

\hline

\mathbf I &  \mathbf I &  \mathbf A &  \mathbf B &  \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C \\
\mathbf A &  \mathbf A &  \mathbf I & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I &  \mathbf C &  \mathbf B \\
\mathbf B &  \mathbf B &  \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C &  \mathbf I &  \mathbf A \\
\mathbf C &  \mathbf C &  \mathbf B &  \mathbf A &  \mathbf I & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I \\

-\mathbf I & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C & \mathbf I & \mathbf A & \mathbf B & \mathbf C \\ -\mathbf A & -\mathbf A & -\mathbf I & \mathbf C & \mathbf B & \mathbf A & \mathbf I & -\mathbf C & -\mathbf B \\ -\mathbf B & -\mathbf B & -\mathbf C & \mathbf I & \mathbf A & \mathbf B & -\mathbf C & -\mathbf I & -\mathbf A \\ -\mathbf C & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I & \mathbf C & \mathbf B & \mathbf A & \mathbf I \end{array}$

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