W
          
          
            24
          
        
      
    
    {\displaystyle \mathrm {W} _{24}}
   generates octads from code words of Hamming weight 8 from the extended binary Golay code, one of five different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24.
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.