Fano plane

E262444
(7,3,1)-design Steiner system finite geometry finite projective plane projective plane of order 2 symmetric block design

The Fano plane is the smallest finite projective plane, consisting of seven points and seven lines with rich symmetrical and combinatorial properties.

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All labels observed (6)

Label Occurrences
Fano matroid 1
Fano plane canonical 1
Fano plane incidence structure 1

Statements (50)

Predicate Object
instanceOf (7,3,1)-design ⓘ
Steiner system ⓘ
finite geometry ⓘ
finite projective plane ⓘ
projective plane of order 2 ⓘ
symmetric block design ⓘ
hasAssociatedMatroid Fano plane self-linksurface differs ⓘ
surface form: Fano matroid
hasAutomorphismGroup PSL(2,7) ⓘ
surface form: PGL(3,2)
hasAutomorphismGroupOrder 168 ⓘ
hasBlockSize 3 ⓘ
hasChromaticNumberOfPointGraph 3 ⓘ
hasCliqueNumberOfPointGraph 3 ⓘ
hasCollineationGroup PSL(2,7) ⓘ
hasCollineationGroupOrder 168 ⓘ
hasDualStructure isomorphic to itself ⓘ
hasGirth 3 ⓘ
hasIncidenceStructure 7 points and 7 lines with 21 incidences ⓘ
hasIndependenceNumberOfPointGraph 3 ⓘ
hasLineGraph isomorphic to its point graph ⓘ
hasLineSetSize 7 ⓘ
hasLinesThroughEachPoint 3 ⓘ
hasNameOrigin Fano plane self-linksurface differs ⓘ
surface form: named after Gino Fano
hasNumberOfLines 7 ⓘ
hasNumberOfPoints 7 ⓘ
hasOrder 2 ⓘ
hasParameterB 7 ⓘ
hasParameterK 3 ⓘ
hasParameterLambda 1 ⓘ
hasParameterR 3 ⓘ
hasParameterV 7 ⓘ
hasPointGraph Fano plane self-linksurface differs ⓘ
surface form: Paley graph of order 7
hasPointSetSize 7 ⓘ
hasPointsPerLine 3 ⓘ
hasReplicationNumber 3 ⓘ
hasSymmetryProperty flag-transitive ⓘ
line-transitive ⓘ
point-transitive ⓘ
isIsomorphicTo projective plane over GF(2) ⓘ
isNotRepresentableOver the real numbers as straight lines in the Euclidean plane ⓘ
isRepresentableOver GF(2) ⓘ
isSelfDual true ⓘ
isSmallest finite projective plane ⓘ
isUsedIn coding theory ⓘ
combinatorial constructions ⓘ
design theory ⓘ
finite geometry ⓘ
matroid theory ⓘ
satisfiesAxiom any two distinct lines meet in a unique point ⓘ
any two distinct points lie on a unique line ⓘ
there exist four points no three of which are collinear ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Klein quartic → relatedTo → Fano plane ⓘ
PSL(2,7) → hasCayleyGraphRelatedTo → Fano plane ⓘ
this entity surface form: Heawood graph
PSL(2,7) → isAutomorphismGroupOf → Fano plane ⓘ
this entity surface form: Fano plane incidence structure
Fano plane → hasPointGraph → Fano plane self-linksurface differs ⓘ
this entity surface form: Paley graph of order 7
Fano plane → hasAssociatedMatroid → Fano plane self-linksurface differs ⓘ
this entity surface form: Fano matroid
Fano plane → hasNameOrigin → Fano plane self-linksurface differs ⓘ
this entity surface form: named after Gino Fano