[[Abstract projective plane]]
# Fano plane

The **Fano plane** is an [[Abstract projective plane]] and [[Galois geometry]] $\mathrm{PG}(2,2) = \mathrm{P}(\mathbb{Z}_{2}^3)$. #m/def/geo/fin 
In addition it forms the unique [[Steiner system]] $\opn S(2,3,7)$, where lines are triads.

![[Fano_plane_nimbers.svg#invert|300]]

The Fano plane is shares an intimate relationship with the [[Binary 7,4,3 Hamming code]].

## Properties

- $\opn{PGL}(3,2) \cong \opn{PGL}(1,7)$[^1988]

  [^1988]: 1988\. [[Sources/@lordGeometryMathieuGroups1988|Geometry of the Mathieu groups and Golay codes]]
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