Group Actions · Hard problem

Label the vertices of a regular heptagon $0,1,\dots,6$. $D_7$ (order $14$) acts on the $7$ diagonals joining vertex $i$ to vertex $i+2\pmod7$ (e.g. $\{0,2\},\{1,3\},\dots$). Find the order of the stabilizer of the diagonal $\{0,2\}$.