<s>[INST] How many combinations of connected midpoints for a regular hexagon? Board game designer here looking for some help with tile design for a hex-tile based game. any help with my image example or wording to make this question more clear is greatly appreciated.
Consider the regular hexagon ABCDEF with midpoints 123456

How many possible groups are there to draw exactly three line segments that connect two midpoints. Example line segments 1-2, 3-4, 5-6 are one group and 1-3,2-4, 5-6 are a separate group. I want the set of all groups that meet the following conditions:
1: Each midpoint is connected by a line segment to exactly one other midpoint
2: Each set is not a rotation of another set (the part I'm really having difficulty with, and I am genuinely curious if this is something calculable.)
So, by the non-mathematical powers of brute force, I compiled the following list of groups:
(1-2, 3-4, 5-6),
(1-2, 3-5, 4-6),
(1-2, 3-6, 4-5),
(1-3, 2-4, 5-6),
(1-3, 2-5, 4-6),
(1-3, 2-6, 4-5),
(1-4, 2-3, 5-6),
(1-4, 2-5, 3-6),
(1-4, 2-6, 3-5),
(1-5, 2-3, 4-6),
(1-5, 2-4, 3-6),
(1-5, 2-6, 3-4),
(1-6, 2-3, 4-5),
(1-6, 2-4, 3-5),
(1-6, 2-5, 3-4)

After that I figured each new group would simply be a rotation of one of the existing groups turned 120 degrees.
So what I need is simply someone to check my work and see if I missed something, and help me root out duplicates. for example (1-2, 3-4, 5-6) and (1-6, 2-3, 4-5) are duplicates if one is rotated 120 degrees.
 [/INST] I doubt there are more than five up to rotation: three with an adjacent pair and two without

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