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# Polytope of Type {3,6}

Atlas Canonical Name : {3,6}*144
Also Known As : {3,6}(2,2)if this polytope has another name.
Group : SmallGroup(144,183)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 12, 36, 24
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Toroidal
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{3,6,2} of size 288
{3,6,4} of size 576
{3,6,3} of size 720
{3,6,6} of size 864
{3,6,4} of size 1152
{3,6,8} of size 1152
{3,6,4} of size 1152
{3,6,6} of size 1440
{3,6,10} of size 1440
{3,6,12} of size 1728
Vertex Figure Of :
{2,3,6} of size 288
{4,3,6} of size 576
{3,3,6} of size 720
{6,3,6} of size 864
{4,3,6} of size 1152
{4,3,6} of size 1152
{6,3,6} of size 1440
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,6}*48
4-fold quotients : {3,6}*36
6-fold quotients : {3,3}*24
12-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,12}*288, {6,6}*288b
3-fold covers : {9,6}*432, {3,6}*432
4-fold covers : {3,6}*576, {12,6}*576a, {6,12}*576c, {6,6}*576b, {12,6}*576d, {6,12}*576e, {3,12}*576
5-fold covers : {15,6}*720e
6-fold covers : {9,12}*864, {3,12}*864, {18,6}*864, {6,6}*864a, {6,6}*864c
7-fold covers : {21,6}*1008b
8-fold covers : {3,12}*1152a, {6,6}*1152a, {12,12}*1152e, {12,12}*1152g, {12,6}*1152a, {6,6}*1152d, {6,6}*1152f, {24,6}*1152g, {24,6}*1152i, {12,12}*1152l, {6,24}*1152j, {6,12}*1152e, {12,12}*1152q, {6,24}*1152m, {3,12}*1152b, {3,24}*1152b, {6,12}*1152g, {3,24}*1152c, {6,12}*1152j
9-fold covers : {27,6}*1296, {9,18}*1296a, {9,6}*1296a, {3,6}*1296, {9,6}*1296b, {3,18}*1296a, {9,6}*1296c, {9,6}*1296d
10-fold covers : {15,12}*1440c, {6,30}*1440g, {30,6}*1440h
11-fold covers : {33,6}*1584
12-fold covers : {9,6}*1728, {3,6}*1728, {36,6}*1728a, {18,12}*1728a, {18,6}*1728a, {36,6}*1728c, {18,12}*1728b, {12,6}*1728a, {6,12}*1728c, {6,6}*1728b, {12,6}*1728d, {6,12}*1728e, {9,12}*1728, {3,12}*1728, {6,12}*1728g, {12,6}*1728g, {6,6}*1728f, {6,12}*1728h, {12,6}*1728h
13-fold covers : {39,6}*1872
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11);;
s1 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11);;
s2 := ( 1, 2)( 5, 6)( 9,10);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(12)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11);
s1 := Sym(12)!( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11);
s2 := Sym(12)!( 1, 2)( 5, 6)( 9,10);
poly := sub<Sym(12)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

```
References : None.
to this polytope