Polytope of Type {3,4,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,3}*1152
Also Known As : 24-cell, {3,4,3}. if this polytope has another name.
Group : SmallGroup(1152,157478)
Rank : 4
Schlafli Type : {3,4,3}
Number of vertices, edges, etc : 24, 96, 96, 24
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,4,3}*576
32-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 8)( 6, 7)(11,13)(12,14)(19,24)(20,23);;
s1 := ( 3, 8)( 4, 7)(13,16)(14,15)(19,21)(20,22);;
s2 := ( 3, 4)( 9,22)(10,21)(11,20)(12,19)(13,23)(14,24)(15,18)(16,17);;
s3 := ( 1,10)( 2, 9)( 3,16)( 4,15)( 5,11)( 6,12)( 7,14)( 8,13)(17,18);;
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,
s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(24)!( 5, 8)( 6, 7)(11,13)(12,14)(19,24)(20,23);
s1 := Sym(24)!( 3, 8)( 4, 7)(13,16)(14,15)(19,21)(20,22);
s2 := Sym(24)!( 3, 4)( 9,22)(10,21)(11,20)(12,19)(13,23)(14,24)(15,18)(16,17);
s3 := Sym(24)!( 1,10)( 2, 9)( 3,16)( 4,15)( 5,11)( 6,12)( 7,14)( 8,13)(17,18);
poly := sub<Sym(24)|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, s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2 >;
References : - Schl�fli, L.; Theorie Der Vielfachen Kontinuit�t, Denkschriften Der Schwe\
izerischen Naturforschenden Gesellschaft, 38, pp1�237 (1901)
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