\(c\)-extensions of the Petersen geometry for \(M_{22}\) (Q1289077)

Let \(G\) be a finite group that acts flag-transitively on a diagram geometry of type \({\buildrel 1 \over \circ}{\buildrel c \over{\text{---}}}\buildrel 2 \over \circ\)---\(\buildrel 3 \over \circ{\buildrel P^* \over{\text{---}}} \buildrel 4 \over \circ\) such that the residue of each element of type 1 is the Petersen geometry for the Mathieu group M\(_{22}\). The authors prove that in this case \(G\) is isomorphic to one of the following groups: \(2^{10}\cdot\text{M}_{22}\), \(2^{11}\cdot\text{M}_{22}\), \(2^{10}\cdot\text{Aut(M}_{22})\), \(2^{11}\cdot\text{Aut(M}_{22})\), \(\text{U}_6(2)\), \(2\cdot\text{U}_6(2)\), \(\text{U}_6(2) : 2\), \(2 \cdot\text{U}_6(2) : 2\), or \(\text{M}_{24}\). For the proof the authors determine generators (involutions, in fact) and relations of \(G\) and then apply the Todd-Coxeter algorithm [see \textit{J. Cannon} and \textit{W. Bosma}, Handbook of Magma Functions, Sydney (1993)]. Moreover, the paper contains two examples of parabolic systems with the above diagram, one for the unitary group \(\text{U}_6(2)\) and the other for \(\text{M}_{24}\).