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# Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*108
Also Known As : {3,6}(3,0), {3,6}6if this polytope has another name.
Group : SmallGroup(108,17)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 9, 27, 18
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Toroidal
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{3,6,2} of size 216
{3,6,3} of size 324
{3,6,4} of size 432
{3,6,6} of size 648
{3,6,6} of size 648
{3,6,8} of size 864
{3,6,3} of size 972
{3,6,9} of size 972
{3,6,10} of size 1080
{3,6,12} of size 1296
{3,6,12} of size 1296
{3,6,14} of size 1512
{3,6,15} of size 1620
{3,6,16} of size 1728
{3,6,4} of size 1728
{3,6,6} of size 1944
{3,6,18} of size 1944
{3,6,18} of size 1944
{3,6,6} of size 1944
{3,6,6} of size 1944
{3,6,6} of size 1944
Vertex Figure Of :
{2,3,6} of size 216
{4,3,6} of size 432
{6,3,6} of size 648
{4,3,6} of size 864
{3,3,6} of size 1296
{4,3,6} of size 1296
{8,3,6} of size 1728
{6,3,6} of size 1944
{6,3,6} of size 1944
{6,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,6}*36
9-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6}*216c
3-fold covers : {9,6}*324a, {9,6}*324b, {9,6}*324c, {9,6}*324d, {3,6}*324, {3,18}*324
4-fold covers : {12,6}*432a, {6,12}*432c, {3,6}*432, {3,12}*432
5-fold covers : {15,6}*540
6-fold covers : {18,6}*648a, {18,6}*648c, {18,6}*648d, {18,6}*648e, {6,6}*648d, {6,18}*648h, {6,6}*648e
7-fold covers : {21,6}*756
8-fold covers : {24,6}*864a, {12,12}*864a, {6,24}*864c, {3,12}*864, {3,24}*864, {6,6}*864a, {6,12}*864a
9-fold covers : {9,18}*972a, {3,18}*972a, {9,6}*972a, {9,6}*972b, {9,18}*972b, {9,6}*972c, {9,18}*972c, {9,18}*972d, {9,18}*972e, {27,6}*972a, {9,6}*972d, {9,18}*972f, {9,18}*972g, {9,18}*972h, {9,18}*972i, {9,6}*972e, {9,18}*972j, {27,6}*972b, {27,6}*972c, {3,6}*972, {3,18}*972b
10-fold covers : {6,30}*1080a, {30,6}*1080b
11-fold covers : {33,6}*1188
12-fold covers : {36,6}*1296a, {36,6}*1296c, {36,6}*1296d, {36,6}*1296e, {12,18}*1296d, {12,6}*1296c, {18,12}*1296e, {18,12}*1296f, {18,12}*1296g, {18,12}*1296h, {6,12}*1296d, {6,36}*1296h, {9,6}*1296a, {3,6}*1296, {3,36}*1296, {9,6}*1296b, {3,12}*1296a, {3,18}*1296a, {9,12}*1296a, {9,6}*1296c, {9,12}*1296b, {9,12}*1296c, {9,6}*1296d, {9,12}*1296d, {12,6}*1296h, {6,12}*1296i
13-fold covers : {39,6}*1404
14-fold covers : {6,42}*1512a, {42,6}*1512b
15-fold covers : {45,6}*1620a, {45,6}*1620b, {45,6}*1620c, {45,6}*1620d, {15,6}*1620, {15,18}*1620
16-fold covers : {48,6}*1728a, {12,24}*1728a, {12,12}*1728a, {12,24}*1728b, {24,12}*1728c, {24,12}*1728e, {6,48}*1728c, {3,6}*1728, {3,24}*1728, {12,12}*1728i, {12,6}*1728a, {12,12}*1728m, {6,12}*1728c, {6,24}*1728b, {6,6}*1728b, {6,24}*1728d, {12,6}*1728d, {6,12}*1728e, {6,12}*1728f, {3,12}*1728, {6,6}*1728c
17-fold covers : {51,6}*1836
18-fold covers : {18,18}*1944a, {18,6}*1944a, {6,18}*1944b, {18,6}*1944d, {18,18}*1944f, {18,6}*1944f, {18,18}*1944h, {18,18}*1944l, {18,18}*1944o, {54,6}*1944a, {18,6}*1944h, {18,18}*1944q, {18,18}*1944t, {18,18}*1944u, {18,18}*1944y, {18,6}*1944i, {18,18}*1944ab, {54,6}*1944c, {54,6}*1944e, {6,6}*1944b, {6,18}*1944k, {18,6}*1944m, {6,18}*1944o, {6,6}*1944d, {6,6}*1944e, {18,6}*1944p, {18,6}*1944q, {18,6}*1944r, {6,6}*1944j, {6,18}*1944u
Permutation Representation (GAP) :
```s0 := (2,3)(4,7)(5,6);;
s1 := (1,4)(2,9)(5,8);;
s2 := (4,5)(6,7)(8,9);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope