27 (number)

Mathematics

Including the null-motif, there are 27 distinct hypergraph motifs.

There are exactly twenty-seven straight lines on a smooth cubic surface, which give a basis of the fundamental representation of Lie algebra \mathrm {E_{6}}.

The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional; its automorphism group is the 52-dimensional exceptional Lie algebra \mathrm {F_{4}}.

There are twenty-seven sporadic groups, if the non-strict group of Lie type \mathrm {T} (with an irreducible representation that is twice that of \mathrm {F_{4}} in 104 dimensions) is included.

In Robin's theorem for the Riemann hypothesis, twenty-seven integers fail to hold \sigma(n) for values n \leq 5040, where \gamma is the Euler–Mascheroni constant; this hypothesis is true if and only if this inequality holds for every larger n.

The Clebsch surface has 27 exceptional lines can be defined over the real numbers.

It is possible to arrange 27 vertices and connect them with edges to create the Holt graph.

27 is 33, and therefore, it is the second tetration of 3 (23).

In other fields

• The 27 club refers to the age when many popular music figures died.
• The ASCII escape character is encoded as .

Notes

References

References

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3. Aschbacher, Michael. (1987). "The 27-dimensional module for E₆. I". [[Springer Science+Business Media.
4. {{Cite OEIS. A121737. Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order.
5. Kac, Victor Grigorievich. (1977). "Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras". [[Taylor & Francis]].
6. (2002). "The Octonions". [[American Mathematical Society]].
7. Lubeck, Frank. (2001). "Smallest degrees of representations of exceptional groups of Lie type". [[Taylor & Francis]].
8. (2010). "Polytopes Derived from Sporadic Simple Groups". [[University of Calgary]] Department of Mathematics and Statistics.
9. Axler, Christian. (2023). "On Robin's inequality". [[Springer Science+Business Media.
10. (1984). "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann". [[Journal de Mathématiques Pures et Appliquées]].
11. {{Cite OEIS. A067698. Positive integers such that sigma(n) is greater than or equal to exp(gamma) * n * log(log(n)).