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Polytope of Type {3,6,3}

Atlas Canonical Name : {3,6,3}*324a
Also Known As : 7T4(1,1)(3,0)if this polytope has another name.
Group : SmallGroup(324,39)
Rank : 4
Schlafli Type : {3,6,3}
Number of vertices, edges, etc : 3, 27, 27, 9
Order of s0s1s2s3 : 9
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Locally Toroidal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,6,3,2} of size 648
{3,6,3,4} of size 1296
{3,6,3,6} of size 1944
Vertex Figure Of :
{2,3,6,3} of size 648
{4,3,6,3} of size 1296
{6,3,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,6,3}*108
9-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,6,6}*648b, {6,6,3}*648b
3-fold covers : {3,6,3}*972, {3,6,9}*972a, {9,6,3}*972b
4-fold covers : {3,6,12}*1296a, {12,6,3}*1296b, {6,6,6}*1296b
5-fold covers : {15,6,3}*1620a, {3,6,15}*1620b
6-fold covers : {3,6,6}*1944a, {6,6,3}*1944a, {3,6,18}*1944a, {6,6,9}*1944a, {9,6,6}*1944b, {18,6,3}*1944b, {3,6,6}*1944f, {6,6,3}*1944g
Permutation Representation (GAP) :
```s0 := (2,3)(5,6)(8,9);;
s1 := (2,3)(5,6)(7,8);;
s2 := (4,7)(5,8)(6,9);;
s3 := (1,4)(2,5)(3,6);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References :
1. Theorem 11E7, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambr\ idge University Press, 2002)

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