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# Polytope of Type {3,3,3}

Atlas Canonical Name : {3,3,3}*120
Also Known As : 4-simplex, {3,3,3}. if this polytope has another name.
Group : SmallGroup(120,34)
Rank : 4
Schlafli Type : {3,3,3}
Number of vertices, edges, etc : 5, 10, 10, 5
Order of s0s1s2s3 : 5
Order of s0s1s2s3s2s1 : 3
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,3,3,2} of size 240
{3,3,3,3} of size 720
{3,3,3,6} of size 1440
{3,3,3,4} of size 1920
Vertex Figure Of :
{2,3,3,3} of size 240
{3,3,3,3} of size 720
{6,3,3,3} of size 1440
{4,3,3,3} of size 1920
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,3,6}*240, {3,6,3}*240, {6,3,3}*240
4-fold covers : {3,12,3}*480, {3,6,6}*480, {6,3,6}*480, {6,6,3}*480
6-fold covers : {3,3,6}*720, {3,6,3}*720a, {3,6,3}*720b, {6,3,3}*720
8-fold covers : {3,12,6}*960, {6,12,3}*960, {3,6,12}*960, {12,6,3}*960, {6,6,6}*960
10-fold covers : {3,6,15}*1200, {15,6,3}*1200
12-fold covers : {3,12,3}*1440a, {3,12,3}*1440b, {3,6,6}*1440a, {3,6,6}*1440b, {3,6,6}*1440c, {6,3,6}*1440a, {6,3,6}*1440b, {6,6,3}*1440a, {6,6,3}*1440b, {6,6,3}*1440c
14-fold covers : {3,6,21}*1680, {21,6,3}*1680
16-fold covers : {3,6,24}*1920, {24,6,3}*1920, {6,6,12}*1920, {6,12,6}*1920a, {12,6,6}*1920, {3,12,12}*1920, {12,12,3}*1920, {6,12,6}*1920b, {3,6,6}*1920, {6,6,3}*1920
Permutation Representation (GAP) :
```s0 := (4,5);;
s1 := (3,4);;
s2 := (2,3);;
s3 := (1,2);;
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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(5)!(4,5);
s1 := Sym(5)!(3,4);
s2 := Sym(5)!(2,3);
s3 := Sym(5)!(1,2);
poly := sub<Sym(5)|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 >;

```
References :
1. Schl�fli, L.; Theorie Der Vielfachen Kontinuit�t, Denkschriften Der Schwe\ izerischen Naturforschenden Gesellschaft, 38, pp1�237 (1901)

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