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# Polytope of Type {3,5,3}

Atlas Canonical Name : {3,5,3}*660
Also Known As : 11-cell, {{3,5}5,{5,3}5}. if this polytope has another name.
Group : SmallGroup(660,13)
Rank : 4
Schlafli Type : {3,5,3}
Number of vertices, edges, etc : 11, 55, 55, 11
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 5
Special Properties :
Universal
Locally Projective
Non-Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,5,3,2} of size 1320
Vertex Figure Of :
{2,3,5,3} of size 1320
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,5,6}*1320, {3,10,3}*1320a, {3,10,3}*1320b, {6,5,3}*1320
Permutation Representation (GAP) :
```s0 := ( 3,11)( 4, 7)( 5, 6)( 8,10);;
s1 := ( 3, 4)( 5, 9)( 7, 8)(10,11);;
s2 := ( 2, 9)( 4, 5)( 6, 7)( 8,10);;
s3 := ( 1, 2)( 3, 4)( 7,11)( 8,10);;
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*s1*s2,
s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(11)!( 3,11)( 4, 7)( 5, 6)( 8,10);
s1 := Sym(11)!( 3, 4)( 5, 9)( 7, 8)(10,11);
s2 := Sym(11)!( 2, 9)( 4, 5)( 6, 7)( 8,10);
s3 := Sym(11)!( 1, 2)( 3, 4)( 7,11)( 8,10);
poly := sub<Sym(11)|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*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 >;

```
References :
1. Coxeter, H. S. M.; A Symmetrical Arrangement of Eleven hemi-Icosahedra, A\ nnals of Discrete Mathematics 20 pp103�114 (1984)
2. Gr�nbaum, B.; Regularity of Graphs, Complexes and Designs, in Probl�mes C\ ombinatoires et Th�orie des Graphes, Colloquium Internationale CNRS, Orsay, 26\ 0 pp191�197 (1977)

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