Minimal permutation representation of the simple group \(F_ 5\) (Q1112169)

Let \(F_ 5\) be the finite simple group of Harada of order \(2^{14}\cdot 3^ 6\cdot 5^ 6\cdot 7\cdot 11\cdot 19\). It contains a subgroup H which is isomorphic to the alternating group \(A_{12}\). The authors prove that H is of minimal possible index in \(F_ 5\) and study the permutation representation of \(F_ 5\) on \(F_ 5/H\). Namely, three theorems are proved: Theorem 1. The minimal index of a proper subgroup in \(F_ 5\) is \(1,140,000=| F_ 5:H|\). Each subgroup of this index in \(F_ 5\) is conjugate to H. - The permutation representation of \(F_ 5\) on cosets \(F_ 5/H\) has rank 12 (Theorem 2; it also gives subdegrees and stabilizers of two points). Theorem 3. \(F_ 5\neq HA\) for every proper subgroup A in \(F_ 5\). In particular, \(F_ 5\) does not contain wide subgroups.