The Fano plane is the projective plane over the finite field . It has seven points and seven lines, each line contains three points, and two distinct lines meet in one point. A line-point incidence matrix therefore satisfies , making it invertible over the real numbers and useful for the transplantation theorem.
Adjoin the inverse-transpose automorphism to the collineation group of the Fano plane. Its maximal subgroups other than the collineation group are unordered incident-pair stabilizers, unordered nonincident-pair stabilizers and Sylow 7-normalizers, of orders 16, 12 and 42.
There are 21 incident point-line pairs and 28 nonincident pairs in the Fano plane, with transitive on each type. Orbit-stabilizer theorem gives the two orders as and .
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The Fano plane is a finite projective plane consisting of 7 points and 7 lines, with the property that each line contains exactly 3 points and each point lies on exactly 3 lines. It is the smallest projective plane and serves as a simple example in the study of combinatorial geometry and finite geometries.