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# Polytope of Type {6,3,3}

Atlas Canonical Name : {6,3,3}*240
Also Known As : 3T4(2,0), {{6,3}4,{3,3}}. if this polytope has another name.
Group : SmallGroup(240,189)
Rank : 4
Schlafli Type : {6,3,3}
Number of vertices, edges, etc : 10, 20, 10, 5
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Locally Toroidal
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,3,3,2} of size 480
{6,3,3,3} of size 1440
Vertex Figure Of :
{2,6,3,3} of size 480
{4,6,3,3} of size 960
{3,6,3,3} of size 1440
{6,6,3,3} of size 1440
{8,6,3,3} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,3,3}*120
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,3,6}*480, {6,6,3}*480
3-fold covers : {6,3,3}*720
4-fold covers : {6,12,3}*960, {12,6,3}*960, {6,6,6}*960
6-fold covers : {6,3,6}*1440a, {6,3,6}*1440b, {6,6,3}*1440a, {6,6,3}*1440b, {6,6,3}*1440c
8-fold covers : {24,6,3}*1920, {6,6,12}*1920, {6,12,6}*1920a, {12,6,6}*1920, {12,12,3}*1920, {6,12,6}*1920b
Permutation Representation (GAP) :
```s0 := (4,5);;
s1 := (3,4)(6,7);;
s2 := (2,3)(6,7);;
s3 := (1,2)(6,7);;
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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References :
1. Theorem 11B5, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambr\ idge University Press, 2002)

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