What group might represent the symmetries of these carbon rings?

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SUMMARY

The discussion focuses on the symmetries of carbon rings, specifically corannulene and coronene, and their representation through symmetry groups. It establishes that these carbon structures do not conform to typical dihedral arrangements found in simpler shapes like hexagons or triangles. Instead, each molecule possesses its own unique symmetry group, which is a subgroup of the symmetric group Sym(n), where n represents the number of vertices. This topic is closely tied to the principles of crystallography.

PREREQUISITES
  • Understanding of symmetry groups in chemistry
  • Familiarity with carbon molecular structures, specifically corannulene and coronene
  • Knowledge of the symmetric group notation, Sym(n)
  • Basic principles of crystallography
NEXT STEPS
  • Research the symmetry groups of specific carbon molecules like corannulene and coronene
  • Study the applications of symmetry in crystallography
  • Explore the mathematical foundations of finite groups and their relevance in chemistry
  • Learn about the implications of molecular symmetry on chemical properties and reactions
USEFUL FOR

Chemists, crystallographers, and students of molecular symmetry who are interested in the structural properties of carbon compounds and their symmetry representations.

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The carbon rings in the upper-middle of this page https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/react3.htm such as corannulene or coronene possess symmetries. But, they are not the typical dihedral arrangements of points, like a single hexagon or single pentagon or single equilateral triangle.

So, what group represents the symmetries of tiles of hexagons, or tiles of triangles and so on?
 
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askmathquestions said:
The carbon rings in the upper-middle of this page https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/react3.htm such as corannulene or coronene possess symmetries. But, they are not the typical dihedral arrangements of points, like a single hexagon or single pentagon or single equilateral triangle.

So, what group represents the symmetries of tiles of hexagons, or tiles of triangles and so on?
This cannot be answered in such a generality. Every molecule has its own (not necessarily different) symmetry group. All that can be said is, that if ##n## is the number of vertices, then it is a subgroup of ##\operatorname{Sym}(n),## i.e. a finite group, which is more of a trivial fact than an answer.

This is subject to crystallography.
 
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