Td molecular point group ($T_d$)

The 24-operation molecular point group of an ideal regular tetrahedron, including improper operations. Its irreducible representations are $A_1,A_2,E,T_1,T_2$, with dimensions $1,1,2,3,3$. The polar-vector representation is $T_2$, the axial-vector representation is $T_1$, and symmetric quadratic functions transform as $A_1+E+T_2$.

The 15 Cartesian displacements of tetrahedral methane have characters $(15,0,-1,-1,3)$ on classes $(E,8C_3,3C_2,6S_4,6{\sigma }_d)$. Removing translations $T_2$ and rotations $T_1$ gives $(9,0,1,-1,3)$ and the stated decomposition. There are nine vibrational coordinates in four frequency families.