The 5-cube, made of ten tesseracts and the 5-cell as its vertex figure, is also regular and one of thirty-one uniform 5-polytopes under the Coxeter 
  
    
      
        
          
            B
          
          
            5
          
        
      
    
    {\displaystyle \mathrm {B} _{5}}
   hypercubic group. The demipenteract, with one hundred and twenty cells, is the only fifth-dimensional semiregular polytope, and has the rectified 5-cell as its vertex figure, which is one of only three semiregular 4-polytopes alongside the rectified 600-cell and the snub 24-cell. In the fifth dimension, there are five regular paracompact honeycombs, all with infinite facets and vertex figures.
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.