Equivalence Classes of Bracelet Beads: Red, White, Blue

[linag96]

Each bead on a bracelet with three beads is either red,
white, or blue.
Define the relation R between bracelets as: (B1, B2),
where B1 and B2 are bracelets, belongs to R if and only
if B2 can be obtained from B1 by rotating it or rotating it
and then reflecting it.
What are the equivalence classes of R?

I'm a little lost on how to make these classes, is it just something like
(red, blue, white), (red, white, blue), (blue, red, white), (blue, white, red) etc?
Thank you for your help.

 

[I like Serena]

Each bead on a bracelet with three beads is either red,
white, or blue.
Define the relation R between bracelets as: (B1, B2),
where B1 and B2 are bracelets, belongs to R if and only
if B2 can be obtained from B1 by rotating it or rotating it
and then reflecting it.
What are the equivalence classes of R?

I'm a little lost on how to make these classes, is it just something like
(red, blue, white), (red, white, blue), (blue, red, white), (blue, white, red) etc?
Thank you for your help.

Hi linag96! ;)

Yes it is.
For instance {(red,red,red)} is an equivalence class.
And so is {(red,red,blue), (red,blue,red), (blue,red,red)}.
Which would the other classes be? (Wondering)

Btw, note that each equivalence class corresponds to a unique bracelet.
We are effectively finding the number of unique bracelets.

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button/.style 2 args={%
circle,
minimum size=0.6cm,
top color=#1!40!white,
bottom color=#1!60!black,
draw=#1!90!black,
thick,
general shadow={%
shadow xshift=.4ex, shadow yshift=-.4ex,
opacity=.5, fill=black!50,
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}]
\newcommand{\bracelet}[4]
{%
\draw #1 node[button={#2}{}] (A) {1}
+(1,-0.5) node[button={#3}{}] (B) {2}
+(2,0) node[button={#4}{}] (C) {3};
\draw[bend right] (A) edge (B) (B) edge (C) (C) edge (A);
}

\node at (-1,0) {\LARGE $\{$};
\bracelet{}{red}{red}{red};
\node at (3,0) {\LARGE $\}$};

\node at (-1,-2) {\LARGE $\{$};
\bracelet{(0,-2)}{red}{red}{green};
\bracelet{(3,-2)}{red}{green}{red};
\bracelet{(6,-2)}{green}{red}{red};
\node at (9,-2) {\LARGE $\}$};

\end{tikzpicture}