The authors show that there are infinitely many \(b\) such that a root of the polynomial \(f_b(x)=x^7+x^6+x^5+bx{}^4+(b-2)x^3-5x^2-2x+1\) generates distinct monogenic septimic fields with Galois group \(\text{PSL}(2,7)\). The proof required an accurate argumentation using standard tools of algebraic number theory. This is the first case that a higher degree infinite parametric family of number fields has been considered so closely, especially that the monogenity has been showed. The result will certainly have further important applications.