The 5-simplex has the Coxeter group 
  
    
      
        
          
            A
          
          
            5
          
        
      
    
    {\displaystyle \mathrm {A} _{5}}
   as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group 
  
    
      
        
          
            S
          
          
            6
          
        
      
    
    {\displaystyle \mathrm {S} _{6}}
  , the only finite symmetric group which has an outer automorphism.
There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537. The sum of the first 3 Fermat primes, 3, 5 and 17, yields 25 or 52, while 257 is the 55th prime number. Combinations from these 5 Fermat primes generate 31 polygons with an odd number of sides that can be constructed purely with a compass and straight-edge, which includes the five-sided regular pentagon. Apropos, 31 is also equal to the sum of the maximum number of areas inside a circle that are formed from the sides and diagonals of the first five 
  
    
      
        n
      
    
    {\displaystyle n}
  -sided polygons, and equal to the maximum number of areas formed by a six-sided polygon; per Mosers circle problem.The number 5 is the fifth Fibonacci number, being 2 plus 3. It is the only Fibonacci number that is equal to its position aside from 1, which is both the first and second Fibonacci numbers. Five is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... (OEIS: A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5).