G-parity of an isospin multiplet ($G=\eta (-1{)}^I$)

For integer isospin $I$ and neutral-state charge conjugation value $\eta$, the entire isospin multiplet has $G=\eta (-1{)}^I$. Both charge conjugation and the second-axis half-turn reverse the first and third generators, so their product commutes with the full isospin algebra. Evaluate it on the $M=0$ state and use ladder connectivity to obtain the value for every $M$.

Each pion has G parity minus one, so an $n$-pion state has $G=(-1{)}^n$. In the isospin-symmetric strong interaction, a parent with positive G parity can decay only to an even pion multiplicity, and a negative parent only to an odd multiplicity. The rule does not forbid small isospin-breaking or electromagnetic amplitudes.