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D4h point group
not Abelian, 10(12) irreducible representations
Subgroups: Cs, Ci, C2, C4, D2, D4, C2v, C4v, C2h, C4h, D2h, D2d, S4

Character table
E 2C4 (z) C2 2C'2 2C''2 i 2S4 σh v d linears,
rotations
quadratic
A1g 1 1 1 1 1 1 1 1 1 1 x2+y2, z2
A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz
B1g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2
B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy
Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z
B1u 1 -1 1 1 -1 -1 1 -1 -1 1
B2u 1 -1 1 -1 1 -1 1 -1 1 -1
Eu 2 0 -2 0 0 -2 0 2 0 0 (x, y)

Product table
 A1gA2gB1gB2gEgA1uA2uB1uB2uEu
A1gA1gA2gB1gB2gEgA1uA2uB1uB2uEu
A2gA2gA1gB2gB1gEgA2uA1uB2uB1uEu
B1gB1gB2gA1gA2gEgB1uB2uA1uA2uEu
B2gB2gB1gA2gA1gEgB2uB1uA2uA1uEu
EgEgEgEgEgA1g+A2g+B1g+B2gEuEuEuEuA1u+A2u+B1u+B2u
A1uA1uA2uB1uB2uEuA1gA2gB1gB2gEg
A2uA2uA1uB2uB1uEuA2gA1gB2gB1gEg
B1uB1uB2uA1uA2uEuB1gB2gA1gA2gEg
B2uB2uB1uA2uA1uEuB2gB1gA2gA1gEg
EuEuEuEuEuA1u+A2u+B1u+B2uEgEgEgEgA1g+A2g+B1g+B2g

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