Group Actions · Hard problem

Label the vertices of a regular $9$-gon $0,1,\dots,8$. $D_9$ (order $18$) acts on the $3$ triangles $\{0,3,6\},\{1,4,7\},\{2,5,8\}$ (vertices $3$ apart) by applying each symmetry to every vertex. Find the order of the stabilizer of $\{0,3,6\}$.