A generalized Barker sequence is a finite sequence \{a_{r}\} of complex numbers having absolute value 1, and possessing a correlation function C(\tau) satisfying the constraint |C(\tau)| \leq 1, \tau \neq 0. Classes of transformations leaving |C(\tau)| invariant are exhibited. Constructions for generalized Barker sequences of various lengths and alphabet sizes are given. Sextic Barker sequences are investigated and examples are given for all lengths through thirteen. No theoretical limit to the length of sextic sequences has been found.