This configuration appear as problem 3845 in Crux Mathematicorum. I see it is very beautiful. This configuration are generalization of Pascal theorem and Brianchon theorem:

Consider six points $A_1$, $A_2$, $\cdots$ , $A_6$ on a circle $(O_A)$ and a point $B_1$ on another circle $(O_B)$. Let the circle $(A_iA_{i+1}B_i)$ meet $(O_B)$ again at $B_{i+1}$ for $i=1,\cdots, 5$. Then the four points $A_6$, $A_1$, $B_1$, $B_6$ lie on a circle. Denoting by $(O_i)$ the circle $(A_iA_{i+1}B_{i+1}B_i)$ for $i=1,\cdots,6$, taking subscripts modulo 6. If $(O_1)$ meets $(O_4)$ at two points $C_3, C_6$, $(O_2)$ meets $(O_5)$ at two points $C_4, C_1$ and $(O_3)$ meets $(O_6)$ at two points $C_2, C_5$, then the six points $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ lie on a circle. The configuration in Figure as follows:

My question: what is the symmetric group of this configuration ?