Here are all the gate codes.

1st Gates

Menoetius: AbCT*bCTA
Coeus: CTAb*bTCT
Lelantos: CAbA*AbCA
Styx: bTAC*CATb
Cronus: TCAb*CAbT

2nd Gates

Menoetius: ATbA*CbAb
Coeus: bTCA*ACTb
Lelantos: TACT*AbTb
Styx: TCTA*AbTC
Cronus: bTbA*ACTA

Gate of Kings

ACbT*bCAT