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# Polytope of Type {18,2,3}

Atlas Canonical Name : {18,2,3}*216
if this polytope has a name.
Group : SmallGroup(216,101)
Rank : 4
Schlafli Type : {18,2,3}
Number of vertices, edges, etc : 18, 18, 3, 3
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{18,2,3,2} of size 432
{18,2,3,3} of size 864
{18,2,3,4} of size 864
{18,2,3,6} of size 1296
{18,2,3,4} of size 1728
{18,2,3,6} of size 1728
Vertex Figure Of :
{2,18,2,3} of size 432
{4,18,2,3} of size 864
{4,18,2,3} of size 864
{4,18,2,3} of size 864
{6,18,2,3} of size 1296
{6,18,2,3} of size 1296
{8,18,2,3} of size 1728
{4,18,2,3} of size 1728
{9,18,2,3} of size 1944
{6,18,2,3} of size 1944
{6,18,2,3} of size 1944
{3,18,2,3} of size 1944
{6,18,2,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9,2,3}*108
3-fold quotients : {6,2,3}*72
6-fold quotients : {3,2,3}*36
9-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,2,3}*432, {18,2,6}*432
3-fold covers : {18,2,9}*648, {18,6,3}*648a, {54,2,3}*648, {18,6,3}*648b
4-fold covers : {72,2,3}*864, {36,2,6}*864, {18,2,12}*864, {18,4,6}*864, {18,4,3}*864
5-fold covers : {90,2,3}*1080, {18,2,15}*1080
6-fold covers : {36,2,9}*1296, {36,6,3}*1296a, {108,2,3}*1296, {18,2,18}*1296, {18,6,6}*1296a, {54,2,6}*1296, {36,6,3}*1296b, {18,6,6}*1296b, {18,6,6}*1296c, {18,6,6}*1296e
7-fold covers : {126,2,3}*1512, {18,2,21}*1512
8-fold covers : {144,2,3}*1728, {36,2,12}*1728, {18,4,12}*1728, {36,4,6}*1728, {72,2,6}*1728, {18,2,24}*1728, {18,8,6}*1728, {36,4,3}*1728, {18,8,3}*1728, {18,4,6}*1728a, {18,4,6}*1728b
9-fold covers : {18,6,9}*1944a, {18,2,27}*1944, {54,2,9}*1944, {54,6,3}*1944a, {18,6,3}*1944a, {18,6,3}*1944b, {162,2,3}*1944, {18,6,9}*1944b, {18,6,3}*1944c, {18,6,3}*1944d, {18,6,3}*1944e, {54,6,3}*1944b
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);;
s1 := ( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);;
s2 := (20,21);;
s3 := (19,20);;
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,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(21)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);
s1 := Sym(21)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,13)(10,11)(12,17)(14,15)(16,18);
s2 := Sym(21)!(20,21);
s3 := Sym(21)!(19,20);
poly := sub<Sym(21)|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, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```

to this polytope