Icosahedral symmetry 
  
    
      
        
          
            I
          
          
            h
          
        
      
    
    {\displaystyle \mathrm {I} _{h}}
   is isomorphic to the alternating group on 5 letters 
  
    
      
        
          
            A
          
          
            5
          
        
      
    
    {\displaystyle \mathrm {A} _{5}}
   of order 120, realized by actions on these uniform polyhedron compounds.
A sixth unitary number, if discovered, would have at least nine odd prime factors.Five is conjectured to be the only odd untouchable number, and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.In figurate numbers, 5 is a pentagonal number, with the sequence of pentagonal numbers starting: 1, 5, 12, 22, 35, ...
5 is a centered tetrahedral number: 1, 5, 15, 35, 69, ... Every centered tetrahedral number with an index of 2, 3 or 4 modulo 5 is divisible by 5. 5 is a square pyramidal number: 1, 5, 14, 30, 55, ... The sum of the first four members is 50 while the fifth indexed member in the sequence is 55. 5 is a centered square number: 1, 5, 13, 25, 41, ... The fifth square number or 52 is 25, which features in the proportions of the two smallest (3, 4, 5) and (5, 12, 13) primitive Pythagorean triples.The factorial of five, or 
  
    
      
        5
      
    
    {\displaystyle 5}
  !