Is there a well known name for the algebraic structure consisting of binary necklaces?

I am considering the set of necklaces (ring graphs) containing $n$ nodes where each node can take on one of two colors.

Operations on the set consist of a discrete set of (only clockwise) rotations and toggling between one or more node's color.

For example, let $n = 4$, then the resulting necklace can be visualized as a $2 \times 2$ board with 4 cells.

The set of rotations that can be applied are $\{ 0, 90, 180, 270 \}$ - thus if the top left cell is black and remainder white, then a 90 degree rotation gives the board with the top right cell black and remainder white.

The set of 'toggles' can be thought of as an XOR operations - thus if we had the board above toggled with its 90 degree clockwise rotation we'd end up with a board with the top row colored black and the remainder white.

Is there a well known name for this structure?

(Any knowledge of its orbits would also be of interest.)

• I'm going to guess that you are asking about group actions, perhaps in some restricted setting for "binary necklaces". However the general idea is that the symmetries of a set $S$ of objects (such as binary necklaces) consist of a group $G$ "acting" on the set, modeled as a mapping $\phi:G\times S \to S$. See the related Wikipedia article to determine if this is a topic of interest. – hardmath Aug 12 '17 at 16:33
• @hardmath thanks for the link; I'll read into the subject some more. – GEL Aug 25 '17 at 3:15