What is the symmetry group of this configuration? 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 ?



 A: Observations.


*

*Every automorphism fixes the pair $\{C_n,C_{n+3}\}$ (indicies are taken modulo $6$), as $C_{n+3}$ is the only point sharing $3$ points with $C_n$.  

*Every automorphism fixes the circle in magenta, as it's the only circle with points sharing $3$ circles.

*The action of the automorphism group on the circle in magenta can be described as $C_2 
^3 ⋊S_3$, where the $C_2$ acts by exchanging $C_n$ and $C_{n+3}$, and the $S_3$ acts by permuting the blocks $\{C_1,C_4\},\{C_2,C_5\},\{C_3,C_6\}$.

*Consider the normal subgroup stabilizing all $C_n$'s. 
One symmetry is exchanging $O_A$ with $O_B$ and $A_n$ with $B_n$. If $O_A$ and $O_B$ are both fixed, the only symmetry remains is exchanging $O_n$ with $O_{n+3}$, $A_n$ with $A_{n+3}$ and $B_n$ with $B_{n+3}$. So the structure of the group is $C_2 \times C_2$.
Conclusion.
The whole group can be described as $(C_2 \times C_2)⋊(C_2^3⋊S_3)$. The elements of the subgroup stabilizing all the $C_n$'s and the automorphism mapping $C_n$ to $C_{n+3}$ while stabilizing everything else commutes with all the elements of the group. So the group can also be described as $C_2 \times C_2 \times C_2 \times (C_2^2⋊S_3)$.
