SU(2) isospin 
τ_{x,}τ_{y}, τ_{z} (τ_{+ }τ_{} τ_{z }and _{ }τ^{2}) 
Pairs u,d quarks (originally, neutron and proton) for the construction and properties of hadrons. 
SU(3) favor 
U^{±}, V^{±}, T^{±}, T_{3}, Λ_{8} 
Combines u,d,s for building hadrons. Extension of SU(2) 
SU(3) color 
U^{±}, V^{±}, T^{±}, T_{3}, Λ_{8} 
Generates QCD related to the color charges (R,G,B) 
U(1) charge 
Q 
Overall phase transformation that requires charge confirmation. 
U(1) Baryon # 
B 
Overall phase transformation that requires Baryon number confirmation. 
U(1) Hypercharge 
Y 
U(1) that leads to Electroweak 
Translation 


Rotation 


Boost 


SU(2)_{L }weak isospin 


Parity 


U(1) baryon number 


Charge conjugation 


Time reversal 














Note, all of the tranformations are ultimately some type of phase transformation. This is due to the unitarity requirement for the transforamtions on quantum states. More than one symmetry is required in our complete system. Our representation should be compatible with the symmetries required.