Polytope of Type {4,3,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,3}*384
Also Known As : 4-cube, {4,3,3}. if this polytope has another name.
Group : SmallGroup(384,5602)
Rank : 4
Schlafli Type : {4,3,3}
Number of vertices, edges, etc : 16, 32, 24, 8
Order of s₀s₁s₂s₃ : 8
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,3,3,2} of size 768
   {4,3,3,3} of size 1920
Vertex Figure Of :
   {2,4,3,3} of size 768
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,3,3}*192
   8-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,3,3}*768b, {4,3,6}*768, {4,6,3}*768c
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s2*s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3> of order 2.
      6 facets:
         2 of {4,3}*48
         4 of 2-fold non-regular quotient of {4,3}*48
      8 vertex figures:
         8 of {3,3}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      5 facets:
         2 of {4,3}*24
         3 of {4,3}*48
      8 vertex figures:
         8 of {3,3}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s3*s2*s1*s0*s1*s2*s3> of order 4.
      5 facets:
         3 of 2-fold non-regular quotient of {4,3}*48
         2 of {2,3}*12
      4 vertex figures:
         4 of {3,3}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3> of order 4.
      4 facets:
         2 of {4,3}*24
         2 of 2-fold non-regular quotient of {4,3}*48
      4 vertex figures:
         4 of {3,3}*24

Permutation Representation (GAP) :

s0 := (3,4)(5,6)(7,8);;
s1 := (1,3)(2,4);;
s2 := (3,5)(4,6);;
s3 := (5,7)(6,8);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(8)!(3,4)(5,6)(7,8);
s1 := Sym(8)!(1,3)(2,4);
s2 := Sym(8)!(3,5)(4,6);
s3 := Sym(8)!(5,7)(6,8);
poly := sub<Sym(8)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1 >;

References :

Schl�fli, L.; Theorie Der Vielfachen Kontinuit�t, Denkschriften Der Schweizerischen Naturforschenden Gesellschaft, 38, pp1�237 (1901)

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