Symmetries
A permutation of the seven points of the Fano plane that carries collinear points (points on the same line) to collinear points (in other words, it "preserves collinearity") is called a "collineation", "automorphism", or "symmetry" of the plane. The full collineation group (or automorphism group, or symmetry group) is the projective linear group PGL(3,2) which in this case is isomorphic to the projective special linear group PSL(2,7) = PSL(3,2), and the general linear group GL(3,2) (which is equal to PGL(3,2) because the field has only one nonzero element). It consists of 168 different permutations.
The automorphism group is made up of 6 conjugacy classes.
All cycle structures except the 7-cycle uniquely define a conjugacy class:
- The identity permutation
- 21 permutations with two 2-cycles
- 42 permutations with a 4-cycle and a 2-cycle
- 56 permutations with two 3-cycles
The 48 permutations with a complete 7-cycle form two distinct conjugacy classes with 24 elements:
- A maps to B, B to C, C to D. Then D is on the same line as A and B.
- A maps to B, B to C, C to D. Then D is on the same line as A and C.
See Fano plane collineations for a complete list.
Hence, by the Pólya enumeration theorem, the number of inequivalent colorings of the Fano plane with n colors is:
Read more about this topic: Fano Plane