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# Polytope of Type {9,2,3}

Atlas Canonical Name : {9,2,3}*108
if this polytope has a name.
Group : SmallGroup(108,16)
Rank : 4
Schlafli Type : {9,2,3}
Number of vertices, edges, etc : 9, 9, 3, 3
Order of s0s1s2s3 : 9
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{9,2,3,2} of size 216
{9,2,3,3} of size 432
{9,2,3,4} of size 432
{9,2,3,6} of size 648
{9,2,3,4} of size 864
{9,2,3,6} of size 864
{9,2,3,5} of size 1080
{9,2,3,8} of size 1728
{9,2,3,12} of size 1728
{9,2,3,6} of size 1944
Vertex Figure Of :
{2,9,2,3} of size 216
{4,9,2,3} of size 432
{6,9,2,3} of size 648
{4,9,2,3} of size 864
{8,9,2,3} of size 1728
{18,9,2,3} of size 1944
{6,9,2,3} of size 1944
{6,9,2,3} of size 1944
{6,9,2,3} of size 1944
{6,9,2,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
2-fold covers : {9,2,6}*216, {18,2,3}*216
3-fold covers : {9,2,9}*324, {9,6,3}*324, {27,2,3}*324
4-fold covers : {36,2,3}*432, {9,2,12}*432, {18,2,6}*432
5-fold covers : {45,2,3}*540, {9,2,15}*540
6-fold covers : {9,2,18}*648, {18,2,9}*648, {9,6,6}*648a, {18,6,3}*648a, {27,2,6}*648, {54,2,3}*648, {9,6,6}*648b, {18,6,3}*648b
7-fold covers : {63,2,3}*756, {9,2,21}*756
8-fold covers : {72,2,3}*864, {9,2,24}*864, {36,2,6}*864, {18,2,12}*864, {18,4,6}*864, {18,4,3}*864, {9,4,6}*864
9-fold covers : {9,6,9}*972, {9,2,27}*972, {27,2,9}*972, {27,6,3}*972, {9,6,3}*972a, {9,6,3}*972b, {81,2,3}*972
10-fold covers : {45,2,6}*1080, {90,2,3}*1080, {9,2,30}*1080, {18,2,15}*1080
11-fold covers : {99,2,3}*1188, {9,2,33}*1188
12-fold covers : {9,2,36}*1296, {36,2,9}*1296, {9,6,12}*1296a, {36,6,3}*1296a, {27,2,12}*1296, {108,2,3}*1296, {18,2,18}*1296, {18,6,6}*1296a, {54,2,6}*1296, {36,6,3}*1296b, {9,6,12}*1296b, {18,6,6}*1296b, {18,6,6}*1296c, {18,6,6}*1296e
13-fold covers : {117,2,3}*1404, {9,2,39}*1404
14-fold covers : {63,2,6}*1512, {126,2,3}*1512, {9,2,42}*1512, {18,2,21}*1512
15-fold covers : {9,2,45}*1620, {45,2,9}*1620, {45,6,3}*1620, {9,6,15}*1620, {135,2,3}*1620, {27,2,15}*1620
16-fold covers : {144,2,3}*1728, {9,2,48}*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, {9,4,12}*1728, {9,8,6}*1728, {9,4,3}*1728, {18,4,6}*1728a, {18,4,6}*1728b
17-fold covers : {153,2,3}*1836, {9,2,51}*1836
18-fold covers : {9,6,18}*1944a, {18,6,9}*1944a, {9,2,54}*1944, {18,2,27}*1944, {27,2,18}*1944, {54,2,9}*1944, {27,6,6}*1944a, {54,6,3}*1944a, {9,6,6}*1944a, {18,6,3}*1944a, {9,6,6}*1944b, {18,6,3}*1944b, {81,2,6}*1944, {162,2,3}*1944, {9,6,18}*1944b, {9,18,6}*1944, {18,6,9}*1944b, {9,6,6}*1944c, {9,6,6}*1944d, {18,6,3}*1944c, {18,6,3}*1944d, {9,6,6}*1944e, {18,6,3}*1944e, {27,6,6}*1944b, {54,6,3}*1944b
Permutation Representation (GAP) :
```s0 := (2,3)(4,5)(6,7)(8,9);;
s1 := (1,2)(3,4)(5,6)(7,8);;
s2 := (11,12);;
s3 := (10,11);;
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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(12)!(2,3)(4,5)(6,7)(8,9);
s1 := Sym(12)!(1,2)(3,4)(5,6)(7,8);
s2 := Sym(12)!(11,12);
s3 := Sym(12)!(10,11);
poly := sub<Sym(12)|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 >;

```

to this polytope