Modular maximal-cyclic groups M_n(2)

A modular maximal-cyclic group group M_n(2) is a semidirect product C_2^n-1⋊C₂, as is the dihedral group, except that C₂ acts on C_2^n-1 as x↦x^{2^n-2+1} instead of x↦x^-1. See Extensions - example.