Polytope of Type {3,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,3,4}*384
Also Known As : 4-cross-polytope, {3,3,4}. if this polytope has another name.
Group : SmallGroup(384,5602)
Rank : 4
Schlafli Type : {3,3,4}
Number of vertices, edges, etc : 8, 24, 32, 16
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 4
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,3,4,2} of size 768
Vertex Figure Of :
   {2,3,3,4} of size 768
   {3,3,3,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,3,4}*192
   8-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,3,8}*768b, {3,6,4}*768c, {6,3,4}*768
Permutation Representation (GAP) : 
s0 := (3,5)(4,6);;
s1 := (5,7)(6,8);;
s2 := (1,7)(2,8);;
s3 := (3,4)(5,6)(7,8);;
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, s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(8)!(3,5)(4,6);
s1 := Sym(8)!(5,7)(6,8);
s2 := Sym(8)!(1,7)(2,8);
s3 := Sym(8)!(3,4)(5,6)(7,8);
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, 
s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3 >;
References :
  1. Schl�fli, L.; Theorie Der Vielfachen Kontinuit�t, Denkschriften Der Schwe\ izerischen Naturforschenden Gesellschaft, 38, pp1�237 (1901)

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