e
            
          
          
            8
          
        
      
    
    {\displaystyle {\mathfrak {e}}_{8}}
  , the largest of all five exceptional groups, also contains the other four as subgroups and is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the 600-cell. There are also five solvable groups that are excluded from finite simple groups of Lie type. The five Mathieu groups constitute the first generation in the happy family of sporadic groups.
In particular, 
  
    
      
        
          
            M
          
          
            11
          
        
      
    
    {\displaystyle \mathrm {M} _{11}}
  , the smallest of all sporadic groups, has a rank 3 action on fifty-five points from an induced action on unordered pairs, as well as two five-dimensional faithful complex irreducible representations over the field with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with n elements.