О регулярных политопах ранга 3

It is proved that if a finite group G is generated by three involutions α, β and γ, such that α and γ commute, and the orders of the products αβ and βγ are greater than 2, then the generating set fα, β, γg makes G the automorphism group of a regular 3-polytope if and only if the intersection ⟨αβ⟩ \ ⟨βγ⟩ contains no non-trivial normal subgroup of G, and the intersection ⟨α, β⟩ \ ⟨β, γ⟩ is not an elementary abelian subgroup of order 4. This criterion complements a theorem by M. Conder and D. Oliveros (J. Combin. Theory Ser. A, 2013, v. 120, no. 6, pp. 1291–1304