In practice, every multicopter (including 6-propeller hexacopter and 4-propeller quadcopter) that I've seen so far has 4 degrees of freedom: pitch, roll, yaw, and climb.
The minimum number of rigid rotors to get 4 degrees of freedom is 4 rotors.
(Single-rotor helicopters have articulated blades and swashplates, which are more complicated than rigid rotors).
One big advantage of using more rotors than the theoretical minimum number is so that if/when something goes wrong that affects one of the 6 rotors, the remaining rotors can still control all 4 degrees of freedom for a controlled landing.
Such multicopters, like single-rotor helicopters, must go through a "roll -- pull collective -- unroll -- push collective" sequence to move a little bit to the right, since they cannot directly control the "forward" or "right" translation degree of freedom.
The maximum number of degrees of freedom of a rigid object in our 3-dimensional world is 6. One way of naming those 6 degrees of freedom: pitch, roll, yaw, climb, forward, right.
Another way of naming the 6 degrees of freedom: heading, elevation, bank, X, Y, Z.
In principle a multirotor helicopter with 6 or more rigid rotors (hexacopters, octocopters, etc.), if the motors are rigidly mounted at the appropriate tilt(*),
or a tandem rotor helicopter if the articulated blades are controlled appropriately, can directly control all 6 degrees of freedom.
Such a helicopter can move sideways directly by increasing sideways thrust, without tilting.
For example, with a tandem rotor helicopter where the forward rotor turns clockwise as viewed from above,
if the controller drives the swashplates so the front rotor tilts forwards and the back rotor tilts backwards, the helicopter as a whole will accelerate to the right without rotating.