171 (number)

171 (one hundred seventy-one) is the natural number following 170 and preceding 172.

171 is the 18th triangular number^[1] and a Jacobsthal number.^[2]

There are 171 transitive relations on three labeled elements,^[3] and 171 combinatorially distinct ways of subdividing a cuboid by flat cuts into a mesh of tetrahedra, without adding extra vertices.^[4]

The diagonals of a regular decagon meet at 171 points, including both crossings and the vertices of the decagon.^[5]

There are 171 faces and edges in the 57-cell, an abstract 4-polytope with hemi-dodecahedral cells that is its own dual polytope.^[6]

Within moonshine theory of sporadic groups, the friendly giant ${\displaystyle \mathbb {M} }$ is defined as having cyclic groups ⟨ ${\displaystyle m}$ ⟩ that are linked with the function,

${\displaystyle f_{m}(\tau )=q^{-1}+a_{1}q+a_{2}q^{2}+...,{\text{ }}a_{k}}$ ∈ ${\displaystyle \mathbb {Z} ,{\text{ }}q=e^{2\pi i\tau },{\text{ }}\tau >0;}$ where ${\displaystyle q}$ is the character of ${\displaystyle \mathbb {M} }$ at ${\displaystyle m}$.

This generates 171 moonshine groups within ${\displaystyle \mathbb {M} }$ associated with ${\displaystyle f_{m}}$ that are principal moduli for different genus zero congruence groups commensurable with the projective linear group ${\displaystyle \operatorname {PSL_{2}} (\mathbb {Z} )}$.^[7]

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