These are also the first five sporadic groups to have been described, defined as 
  
    
      
        
          
            M
          
          
            n
          
        
      
    
    {\displaystyle \mathrm {M} _{n}}
   multiply transitive permutation groups on 
  
    
      
        n
      
    
    {\displaystyle n}
   objects, with 
  
    
      
        n
      
    
    {\displaystyle n}
   ∈ {11, 12, 22, 23, 24}.
On the other hand, whereas 
  
    
      
        
          
            M
          
          
            11
          
        
      
    
    {\displaystyle \mathrm {M} _{11}}
   is sharply 4-transitive, 
  
    
      
        
          
            M
          
          
            12
          
        
      
    
    {\displaystyle \mathrm {M} _{12}}
   is sharply 5-transitive and 
  
    
      
        
          
            M
          
          
            24
          
        
      
    
    {\displaystyle \mathrm {M} _{24}}
   is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups.