Quaternary direct products for Point Group D4h

Point Group D_4h

Quaternary direct products

**A_1g⊙A_1g⊙**
⊙	A_1g	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
A_1g	A_1g
A_2g	A_2g	A_1g
B_1g	B_1g	B_2g	A_1g
B_2g	B_2g	B_1g	A_2g	A_1g
E_g	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	A_1u	A_2u	B_1u	B_2u	E_u	A_1g
A_2u	A_2u	A_1u	B_2u	B_1u	E_u	A_2g	A_1g
B_1u	B_1u	B_2u	A_1u	A_2u	E_u	B_1g	B_2g	A_1g
B_2u	B_2u	B_1u	A_2u	A_1u	E_u	B_2g	B_1g	A_2g	A_1g
E_u	E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_1g⊙A_2g⊙**
⊙	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
A_2g	A_2g
B_1g	B_1g	A_2g
B_2g	B_2g	A_1g	A_2g
E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	A_1u	B_2u	B_1u	E_u	A_2g
A_2u	A_2u	B_1u	B_2u	E_u	A_1g	A_2g
B_1u	B_1u	A_2u	A_1u	E_u	B_2g	B_1g	A_2g
B_2u	B_2u	A_1u	A_2u	E_u	B_1g	B_2g	A_1g	A_2g
E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_1g⊙B_1g⊙**
⊙	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
B_1g	B_1g
B_2g	B_2g	B_1g
E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	A_1u	A_2u	E_u	B_1g
A_2u	A_2u	A_1u	E_u	B_2g	B_1g
B_1u	B_1u	B_2u	E_u	A_1g	A_2g	B_1g
B_2u	B_2u	B_1u	E_u	A_2g	A_1g	B_2g	B_1g
E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_1g⊙B_2g⊙**
⊙	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
B_2g	B_2g
E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	A_1u	E_u	B_2g
A_2u	A_2u	E_u	B_1g	B_2g
B_1u	B_1u	E_u	A_2g	A_1g	B_2g
B_2u	B_2u	E_u	A_1g	A_2g	B_1g	B_2g
E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_1g⊙E_g⊙**
⊙	E_g	A_1u	A_2u	B_1u	B_2u	E_u
E_g	4E_g
A_1u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g
A_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g
B_1u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g
B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g
E_u	4E_u	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	4E_g

**A_1g⊙A_1u⊙**
⊙	A_1u	A_2u	B_1u	B_2u	E_u
A_1u	A_1u
A_2u	A_2u	A_1u
B_1u	B_1u	B_2u	A_1u
B_2u	B_2u	B_1u	A_2u	A_1u
E_u	E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**A_1g⊙A_2u⊙**
⊙	A_2u	B_1u	B_2u	E_u
A_2u	A_2u
B_1u	B_1u	A_2u
B_2u	B_2u	A_1u	A_2u
E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**A_1g⊙B_1u⊙**
⊙	B_1u	B_2u	E_u
B_1u	B_1u
B_2u	B_2u	B_1u
E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**A_1g⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	B_2u
E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**A_1g⊙E_u⊙**
⊙	E_u
E_u	4E_u

**A_2g⊙A_2g⊙**
⊙	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
A_2g	A_1g
B_1g	B_2g	A_1g
B_2g	B_1g	A_2g	A_1g
E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	A_2u	B_1u	B_2u	E_u	A_1g
A_2u	A_1u	B_2u	B_1u	E_u	A_2g	A_1g
B_1u	B_2u	A_1u	A_2u	E_u	B_1g	B_2g	A_1g
B_2u	B_1u	A_2u	A_1u	E_u	B_2g	B_1g	A_2g	A_1g
E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_2g⊙B_1g⊙**
⊙	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
B_1g	B_2g
B_2g	B_1g	B_2g
E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	A_2u	A_1u	E_u	B_2g
A_2u	A_1u	A_2u	E_u	B_1g	B_2g
B_1u	B_2u	B_1u	E_u	A_2g	A_1g	B_2g
B_2u	B_1u	B_2u	E_u	A_1g	A_2g	B_1g	B_2g
E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_2g⊙B_2g⊙**
⊙	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
B_2g	B_1g
E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	A_2u	E_u	B_1g
A_2u	A_1u	E_u	B_2g	B_1g
B_1u	B_2u	E_u	A_1g	A_2g	B_1g
B_2u	B_1u	E_u	A_2g	A_1g	B_2g	B_1g
E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_2g⊙E_g⊙**
⊙	E_g	A_1u	A_2u	B_1u	B_2u	E_u
E_g	4E_g
A_1u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g
A_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g
B_1u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g
B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g
E_u	4E_u	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	4E_g

**A_2g⊙A_1u⊙**
⊙	A_1u	A_2u	B_1u	B_2u	E_u
A_1u	A_2u
A_2u	A_1u	A_2u
B_1u	B_2u	B_1u	A_2u
B_2u	B_1u	B_2u	A_1u	A_2u
E_u	E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**A_2g⊙A_2u⊙**
⊙	A_2u	B_1u	B_2u	E_u
A_2u	A_1u
B_1u	B_2u	A_1u
B_2u	B_1u	A_2u	A_1u
E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**A_2g⊙B_1u⊙**
⊙	B_1u	B_2u	E_u
B_1u	B_2u
B_2u	B_1u	B_2u
E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**A_2g⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	B_1u
E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**A_2g⊙E_u⊙**
⊙	E_u
E_u	4E_u

**B_1g⊙B_1g⊙**
⊙	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
B_1g	A_1g
B_2g	A_2g	A_1g
E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	B_1u	B_2u	E_u	A_1g
A_2u	B_2u	B_1u	E_u	A_2g	A_1g
B_1u	A_1u	A_2u	E_u	B_1g	B_2g	A_1g
B_2u	A_2u	A_1u	E_u	B_2g	B_1g	A_2g	A_1g
E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**B_1g⊙B_2g⊙**
⊙	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
B_2g	A_2g
E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	B_1u	E_u	A_2g
A_2u	B_2u	E_u	A_1g	A_2g
B_1u	A_1u	E_u	B_2g	B_1g	A_2g
B_2u	A_2u	E_u	B_1g	B_2g	A_1g	A_2g
E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**B_1g⊙E_g⊙**
⊙	E_g	A_1u	A_2u	B_1u	B_2u	E_u
E_g	4E_g
A_1u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g
A_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g
B_1u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g
B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g
E_u	4E_u	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	4E_g

**B_1g⊙A_1u⊙**
⊙	A_1u	A_2u	B_1u	B_2u	E_u
A_1u	B_1u
A_2u	B_2u	B_1u
B_1u	A_1u	A_2u	B_1u
B_2u	A_2u	A_1u	B_2u	B_1u
E_u	E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**B_1g⊙A_2u⊙**
⊙	A_2u	B_1u	B_2u	E_u
A_2u	B_2u
B_1u	A_1u	B_2u
B_2u	A_2u	B_1u	B_2u
E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**B_1g⊙B_1u⊙**
⊙	B_1u	B_2u	E_u
B_1u	A_1u
B_2u	A_2u	A_1u
E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**B_1g⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	A_2u
E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**B_1g⊙E_u⊙**
⊙	E_u
E_u	4E_u

**B_2g⊙B_2g⊙**
⊙	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
B_2g	A_1g
E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g
A_1u	B_2u	E_u	A_1g
A_2u	B_1u	E_u	A_2g	A_1g
B_1u	A_2u	E_u	B_1g	B_2g	A_1g
B_2u	A_1u	E_u	B_2g	B_1g	A_2g	A_1g
E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**B_2g⊙E_g⊙**
⊙	E_g	A_1u	A_2u	B_1u	B_2u	E_u
E_g	4E_g
A_1u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g
A_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g
B_1u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g
B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	E_g	E_g	E_g	E_g
E_u	4E_u	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	4E_g

**B_2g⊙A_1u⊙**
⊙	A_1u	A_2u	B_1u	B_2u	E_u
A_1u	B_2u
A_2u	B_1u	B_2u
B_1u	A_2u	A_1u	B_2u
B_2u	A_1u	A_2u	B_1u	B_2u
E_u	E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**B_2g⊙A_2u⊙**
⊙	A_2u	B_1u	B_2u	E_u
A_2u	B_1u
B_1u	A_2u	B_1u
B_2u	A_1u	B_2u	B_1u
E_u	E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**B_2g⊙B_1u⊙**
⊙	B_1u	B_2u	E_u
B_1u	A_2u
B_2u	A_1u	A_2u
E_u	E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**B_2g⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	A_1u
E_u	E_u	A_1u⊕A_2u⊕B_1u⊕B_2u

**B_2g⊙E_u⊙**
⊙	E_u
E_u	4E_u

**E_g⊙E_g⊙**
⊙	E_g	A_1u	A_2u	B_1u	B_2u	E_u
E_g	4A_1g⊕4A_2g⊕4B_1g⊕4B_2g
A_1u	4E_u	A_1g⊕A_2g⊕B_1g⊕B_2g
A_2u	4E_u	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g
B_1u	4E_u	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g
B_2u	4E_u	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g	A_1g⊕A_2g⊕B_1g⊕B_2g
E_u	4A_1u⊕4A_2u⊕4B_1u⊕4B_2u	4E_g	4E_g	4E_g	4E_g	4A_1g⊕4A_2g⊕4B_1g⊕4B_2g

**E_g⊙A_1u⊙**
⊙	A_1u	A_2u	B_1u	B_2u	E_u
A_1u	E_u
A_2u	E_u	E_u
B_1u	E_u	E_u	E_u
B_2u	E_u	E_u	E_u	E_u
E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	4E_u

**E_g⊙A_2u⊙**
⊙	A_2u	B_1u	B_2u	E_u
A_2u	E_u
B_1u	E_u	E_u
B_2u	E_u	E_u	E_u
E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	4E_u

**E_g⊙B_1u⊙**
⊙	B_1u	B_2u	E_u
B_1u	E_u
B_2u	E_u	E_u
E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	A_1u⊕A_2u⊕B_1u⊕B_2u	4E_u

**E_g⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	E_u
E_u	A_1u⊕A_2u⊕B_1u⊕B_2u	4E_u

**E_g⊙E_u⊙**
⊙	E_u
E_u	4A_1u⊕4A_2u⊕4B_1u⊕4B_2u

**A_1u⊙A_1u⊙**
⊙	A_1u	A_2u	B_1u	B_2u	E_u
A_1u	A_1g
A_2u	A_2g	A_1g
B_1u	B_1g	B_2g	A_1g
B_2u	B_2g	B_1g	A_2g	A_1g
E_u	E_g	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_1u⊙A_2u⊙**
⊙	A_2u	B_1u	B_2u	E_u
A_2u	A_2g
B_1u	B_1g	A_2g
B_2u	B_2g	A_1g	A_2g
E_u	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_1u⊙B_1u⊙**
⊙	B_1u	B_2u	E_u
B_1u	B_1g
B_2u	B_2g	B_1g
E_u	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_1u⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	B_2g
E_u	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_1u⊙E_u⊙**
⊙	E_u
E_u	4E_g

**A_2u⊙A_2u⊙**
⊙	A_2u	B_1u	B_2u	E_u
A_2u	A_1g
B_1u	B_2g	A_1g
B_2u	B_1g	A_2g	A_1g
E_u	E_g	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_2u⊙B_1u⊙**
⊙	B_1u	B_2u	E_u
B_1u	B_2g
B_2u	B_1g	B_2g
E_u	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_2u⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	B_1g
E_u	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**A_2u⊙E_u⊙**
⊙	E_u
E_u	4E_g

**B_1u⊙B_1u⊙**
⊙	B_1u	B_2u	E_u
B_1u	A_1g
B_2u	A_2g	A_1g
E_u	E_g	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**B_1u⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	A_2g
E_u	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**B_1u⊙E_u⊙**
⊙	E_u
E_u	4E_g

**B_2u⊙B_2u⊙**
⊙	B_2u	E_u
B_2u	A_1g
E_u	E_g	A_1g⊕A_2g⊕B_1g⊕B_2g

**B_2u⊙E_u⊙**
⊙	E_u
E_u	4E_g

**E_u⊙E_u⊙**
⊙	E_u
E_u	4A_1g⊕4A_2g⊕4B_1g⊕4B_2g