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- We can have up to 3 symbols, which I suspect is because we're in 3D space. I'd love a more complete explanation of this if you've got one. What I don't understand is why we ever need three. As far as I can tell, the first two symbols will unambiguously imply the third. Is this just a matter of convention? (The only exception here is when 2/m 2/m 2/m is sometimes written as mmm which is ambiguous with mm2 if you only give the first two symbols.) I'll assume it is convention in the following.
- When there's only one symbol, it just means there's some rotation/rotoinversion axis which can arbitrarily be defined as the z axis.
- If there is more than one symbol and they're all of order-2 then
- A third order-2 axis is generated by two order-2 axes, so there must be 3 symbols, and
- The condition of closure of a point group means all three axes must be orthogonal, and convention is to put x, y, z axes in order.

- If there is a higher (n
^{th}) order axis then convention dictates this symbol goes first, and following this:- For odd n there are n equivalent axes which are referred to by a second symbol.
- For even n, the second symbol will represent n/2 equivalent directions (the secondary directions). The operators along the secondary directions generate another order-2 rotation or mirror plane along the n/2 directions bisecting the secondary directions. The third symbol represents these tertiary directions. The bisection is required by the closure of the group.

- For multiple higher order axis, things get complicated, and I don't fully understand how the notation is consistent in these cases. For the two cases, 23 and m3, where the notation might imply only one higher order axis, I guess the fact that the higher order comes second indicates that there are multiple of them? The Wikipedia page implies that these crystallographic point groups are always those of a cubic crystal system. I can't quite see myself to an explanation of this fact. Accepting that, however, I appreciate that the three symbols refer to the equivalent x, y and z axes, the body diagonals and the face diagonals respectively.