There are five exceptional Lie groups: 
  
    
      
        
          
            
              g
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathfrak {g}}_{2}}
  , 
  
    
      
        
          
            
              f
            
          
          
            4
          
        
      
    
    {\displaystyle {\mathfrak {f}}_{4}}
  , 
  
    
      
        
          
            
              e
            
          
          
            6
          
        
      
    
    {\displaystyle {\mathfrak {e}}_{6}}
  , 
  
    
      
        
          
            
              e
            
          
          
            7
          
        
      
    
    {\displaystyle {\mathfrak {e}}_{7}}
  , and 
  
    
      
        
          
            
              e
            
          
          
            8
          
        
      
    
    {\displaystyle {\mathfrak {e}}_{8}}
  .
The 5-cell, or pentatope, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry 
  
    
      
        
          
            A
          
          
            4
          
        
      
    
    {\displaystyle \mathrm {A} _{4}}
   of order 120 = 5! and 
  
    
      
        
          
            S
          
          
            5
          
        
      
    
    {\displaystyle \mathrm {S} _{5}}
   group structure.