  We give the formula for the maximal systole of the surface admits the largest
$S^3$-extendable abelian group symmetry. The result we get is
$2\mathrm{arccosh} K$. Here \begin{eqnarray*}
  K &=& \sqrt[3]{\frac{1}{216}L^3 +\frac{1}{8} L^2 + \frac{5}{8} L -
\frac{1}{8} + \sqrt{\frac{1}{108}L(L^2+18L+27)} }
  & & + \sqrt[3]{\frac{1}{216}L^3 +\frac{1}{8} L^2 + \frac{5}{8} L -
\frac{1}{8} - \sqrt{\frac{1}{108}L(L^2+18L+27)} }
  & & + \frac{L+3}{6}. \end{eqnarray*} and $L= 4\cos^2 \frac{\pi}{g-1}$.
