page_title: Fano plane
section_title: Cycle structure
hierarchical_section_title: Fano plane / Collineations / Cycle structure
context_page_description: In finite geometry, the Fano plane is the finite projective plane of order 2. It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. The standard notation for this plane, as a member of a family of projective spaces, is PG where PG stands for "projective geometry", the first parameter is the geometric dimension and the second parameter is the order.
  The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study.
context_section_description: The collineation group, thought of as a permutation group of the 7 points as numbered in the figure, is generated by:
  (1432657), (162)(374), (14)(27), (17)(24), (17)(24)(36).
  It comprises 6 conjugacy classes. 
  The following cycle structures each define a single 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:
caption: A nimber numbering of the Fano plane