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I'm trying to describe Bravais lattice point groups as permutations of lattice points. In doing so, I encounter a problem: I can only find descriptions of Bravais lattice point groups in terms of crystal systems. However, Bravais lattices can be in the same crystal family and have the "same" point group but have different permutation groups for their lattice points.

To make this clear, we can take an example in 2d. As explained in this Wikipedia page, both the rectangular and centered rectangular lattices are part of the same crystal family and can be seen as having the same point group $D_2$. However, upon application of a symmetry transformation points are not permuted in the same way for both lattices.

Another way to see this is by considering a primitive unit cell for each lattice and look at its symmetry. The rectangular unit cell has symmetry elements : identity, 180 rotation, horizontal reflection, vertical reflection. rectangle

For the rhombohedral unit cell, symmetry elements are : identity, 180 rotation, reflection with respect to both diagonals. rhombus

Clearly in both cases, applying a symmetry element will give you different permutations of lattice points and that's what I'm interested in. We could see them as different groups, the operations are different and they are different subgroups of the square.

But since they are isomorphic and can be mapped to each other by choosing the centered rectangular unit cell instead of the rectangular one, they are both labeled as $D_2$ on Wikipedia and any resource I could find.

My question : Is there a classification of Bravais lattice point groups in 3D that actually accounts for these differences? Alternatively, is there a description of the shapes of all primitive cells (that doesn't use base, body or face-centered unit cells)? With that, I could find these groups by myself by looking at the symmetry of each shape, although that would be tedious.

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You are mixing two different concepts, the Bravais lattices and the point groups; the “Bravais lattice point groups” do not exist to my knowledge and I don't understand what you clearly mean with this definition.

There exist 14 3-dimensional Bravais lattices and 32 point groups; by combining Bravais lattices with point groups, the 73 symmorphic space groups are obtained (by considering also non-symmorphic symmetry operations, a total number of 230 space group results).

I would suggest to revise these basic concepts and reformulate, in the case, your question.

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