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What is the number of irreducible representations of the D-4 group.
Find their dimensions and construct the character table for the group.
You may use the notation and information given below.}

  • $\oslash$ Notation for group elements
  • Anticlockwise rotation about \(z\)- axis by angle \(\pi/2\) \(:\to 4_z\).
  • Anticlockwise rotation about \(z\)- axis by angle \(\pi\)\(:\to 4_z^2\).
  • Anticlockwise rotation about \(z\)- axis by angle \(3\pi/2\)\(:\to 4_z^3\).
  • Reflection in a plane perpendicular to \(X\)- axis \(:\to m_x\)
  • Reflection in a plane perpendicular to \(Y\)- axis \(:\to m_y\)
  • Reflection in plane containing \(Z\)- axis and diagonal 13 \(:\to m_{13}\)
  • Reflection in plane containing \(Z\)- axis and diagonal 24 \(:\to m_{24}\)

The classes are given to be  \begin{equation*}  \begin{array}{ll} C_1=\{e\}.&  \\
C_2 =\{4_z^2\} &C_3 =\{4_z, 4_z^3\} \\
C_4=\{m_x, m_y\} &C_5 =\{m_{13}, m_{24}\}
\end{array}

You may use class multiplication rules \begin{equation*}  \begin{array}{llll}
C_2^2=C_1         & C_2C_3=C_3    & C_2C_4=C_4 & C_2C_5 = C_5\\
C_3^2 = 2C_1+C_2  & C_3C_4= 2C_5  & C_3 C_5 = 2C_4 &\\
C_4^2 =2C_1+2C_2  & C_4 C_5= 2C_3  &&\\
C_5^2= 2C_1+2C_2  &&&
\end{array} \end{equation*}

D4 Symmetries of a Square

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4727:Diamond Point

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