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# Polytope of Type {9,2,2}

Atlas Canonical Name : {9,2,2}*72
if this polytope has a name.
Group : SmallGroup(72,17)
Rank : 4
Schlafli Type : {9,2,2}
Number of vertices, edges, etc : 9, 9, 2, 2
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{9,2,2,2} of size 144
{9,2,2,3} of size 216
{9,2,2,4} of size 288
{9,2,2,5} of size 360
{9,2,2,6} of size 432
{9,2,2,7} of size 504
{9,2,2,8} of size 576
{9,2,2,9} of size 648
{9,2,2,10} of size 720
{9,2,2,11} of size 792
{9,2,2,12} of size 864
{9,2,2,13} of size 936
{9,2,2,14} of size 1008
{9,2,2,15} of size 1080
{9,2,2,16} of size 1152
{9,2,2,17} of size 1224
{9,2,2,18} of size 1296
{9,2,2,19} of size 1368
{9,2,2,20} of size 1440
{9,2,2,21} of size 1512
{9,2,2,22} of size 1584
{9,2,2,23} of size 1656
{9,2,2,24} of size 1728
{9,2,2,25} of size 1800
{9,2,2,26} of size 1872
{9,2,2,27} of size 1944
Vertex Figure Of :
{2,9,2,2} of size 144
{4,9,2,2} of size 288
{6,9,2,2} of size 432
{4,9,2,2} of size 576
{8,9,2,2} of size 1152
{18,9,2,2} of size 1296
{6,9,2,2} of size 1296
{6,9,2,2} of size 1296
{6,9,2,2} of size 1296
{6,9,2,2} of size 1296
{6,9,2,2} of size 1728
{12,9,2,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {9,2,4}*144, {18,2,2}*144
3-fold covers : {27,2,2}*216, {9,2,6}*216, {9,6,2}*216
4-fold covers : {9,2,8}*288, {36,2,2}*288, {18,2,4}*288, {18,4,2}*288a, {9,4,2}*288
5-fold covers : {9,2,10}*360, {45,2,2}*360
6-fold covers : {27,2,4}*432, {54,2,2}*432, {9,2,12}*432, {9,6,4}*432, {18,2,6}*432, {18,6,2}*432a, {18,6,2}*432b
7-fold covers : {9,2,14}*504, {63,2,2}*504
8-fold covers : {9,2,16}*576, {36,4,2}*576a, {36,2,4}*576, {18,4,4}*576, {72,2,2}*576, {18,2,8}*576, {18,8,2}*576, {9,4,4}*576b, {9,8,2}*576, {18,4,2}*576
9-fold covers : {81,2,2}*648, {9,2,18}*648, {9,18,2}*648, {9,6,6}*648a, {9,6,2}*648a, {27,2,6}*648, {27,6,2}*648, {9,6,6}*648b
10-fold covers : {9,2,20}*720, {45,2,4}*720, {18,2,10}*720, {18,10,2}*720, {90,2,2}*720
11-fold covers : {9,2,22}*792, {99,2,2}*792
12-fold covers : {27,2,8}*864, {108,2,2}*864, {54,2,4}*864, {54,4,2}*864a, {9,2,24}*864, {9,6,8}*864, {27,4,2}*864, {36,2,6}*864, {36,6,2}*864a, {36,6,2}*864b, {18,2,12}*864, {18,12,2}*864a, {18,4,6}*864, {18,6,4}*864a, {18,6,4}*864b, {18,12,2}*864b, {9,6,2}*864, {9,4,6}*864, {9,12,2}*864
13-fold covers : {9,2,26}*936, {117,2,2}*936
14-fold covers : {9,2,28}*1008, {63,2,4}*1008, {18,2,14}*1008, {18,14,2}*1008, {126,2,2}*1008
15-fold covers : {27,2,10}*1080, {135,2,2}*1080, {9,6,10}*1080, {45,2,6}*1080, {45,6,2}*1080, {9,2,30}*1080
16-fold covers : {9,2,32}*1152, {36,4,4}*1152, {18,4,8}*1152a, {18,8,4}*1152a, {36,8,2}*1152a, {72,4,2}*1152a, {18,4,8}*1152b, {18,8,4}*1152b, {36,8,2}*1152b, {72,4,2}*1152b, {18,4,4}*1152a, {36,4,2}*1152a, {36,2,8}*1152, {72,2,4}*1152, {18,2,16}*1152, {18,16,2}*1152, {144,2,2}*1152, {9,4,4}*1152b, {9,8,2}*1152, {9,8,4}*1152, {9,4,8}*1152, {36,4,2}*1152b, {18,4,4}*1152d, {18,4,2}*1152b, {36,4,2}*1152c, {18,8,2}*1152b, {18,8,2}*1152c
17-fold covers : {9,2,34}*1224, {153,2,2}*1224
18-fold covers : {81,2,4}*1296, {162,2,2}*1296, {9,2,36}*1296, {9,6,12}*1296a, {27,2,12}*1296, {9,18,4}*1296, {9,6,4}*1296a, {27,6,4}*1296, {18,2,18}*1296, {18,18,2}*1296a, {18,18,2}*1296c, {18,6,6}*1296a, {18,6,2}*1296a, {18,6,2}*1296b, {54,2,6}*1296, {54,6,2}*1296a, {54,6,2}*1296b, {9,6,12}*1296b, {9,6,4}*1296e, {18,6,6}*1296b, {18,6,6}*1296c, {18,6,6}*1296d, {18,6,6}*1296e, {18,6,2}*1296i
19-fold covers : {9,2,38}*1368, {171,2,2}*1368
20-fold covers : {9,2,40}*1440, {45,2,8}*1440, {36,2,10}*1440, {36,10,2}*1440, {18,2,20}*1440, {18,20,2}*1440a, {18,4,10}*1440, {18,10,4}*1440, {180,2,2}*1440, {90,2,4}*1440, {90,4,2}*1440a, {9,4,10}*1440, {45,4,2}*1440
21-fold covers : {27,2,14}*1512, {189,2,2}*1512, {9,6,14}*1512, {63,2,6}*1512, {63,6,2}*1512, {9,2,42}*1512
22-fold covers : {9,2,44}*1584, {99,2,4}*1584, {18,2,22}*1584, {18,22,2}*1584, {198,2,2}*1584
23-fold covers : {9,2,46}*1656, {207,2,2}*1656
24-fold covers : {27,2,16}*1728, {108,4,2}*1728a, {108,2,4}*1728, {54,4,4}*1728, {216,2,2}*1728, {54,2,8}*1728, {54,8,2}*1728, {9,2,48}*1728, {9,6,16}*1728, {27,4,4}*1728b, {27,8,2}*1728, {36,2,12}*1728, {36,6,4}*1728a, {18,4,12}*1728, {18,12,4}*1728a, {36,4,6}*1728, {72,2,6}*1728, {72,6,2}*1728a, {72,6,2}*1728b, {18,2,24}*1728, {18,24,2}*1728a, {18,6,8}*1728a, {18,8,6}*1728, {36,12,2}*1728a, {36,12,2}*1728b, {36,6,4}*1728b, {18,6,8}*1728b, {18,24,2}*1728b, {18,12,4}*1728b, {54,4,2}*1728, {9,12,2}*1728, {9,6,4}*1728a, {9,4,12}*1728, {9,24,2}*1728, {9,8,6}*1728, {9,12,4}*1728, {18,4,6}*1728a, {18,6,4}*1728, {18,6,6}*1728, {18,6,2}*1728, {36,6,2}*1728, {18,4,6}*1728b, {18,12,2}*1728a, {18,12,2}*1728b
25-fold covers : {9,2,50}*1800, {225,2,2}*1800, {9,10,2}*1800, {45,2,10}*1800, {45,10,2}*1800
26-fold covers : {9,2,52}*1872, {117,2,4}*1872, {18,2,26}*1872, {18,26,2}*1872, {234,2,2}*1872
27-fold covers : {243,2,2}*1944, {9,6,18}*1944a, {9,18,2}*1944a, {9,2,54}*1944, {27,2,18}*1944, {27,18,2}*1944, {27,6,6}*1944a, {27,6,2}*1944a, {9,6,6}*1944a, {9,6,2}*1944d, {9,18,2}*1944h, {9,18,2}*1944i, {9,6,6}*1944b, {9,6,2}*1944e, {27,6,2}*1944b, {27,6,2}*1944c, {81,2,6}*1944, {81,6,2}*1944, {9,6,18}*1944b, {9,18,6}*1944, {9,6,6}*1944c, {9,6,6}*1944d, {9,6,6}*1944e, {27,6,6}*1944b
Permutation Representation (GAP) :
```s0 := (2,3)(4,5)(6,7)(8,9);;
s1 := (1,2)(3,4)(5,6)(7,8);;
s2 := (10,11);;
s3 := (12,13);;
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, 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(13)!(2,3)(4,5)(6,7)(8,9);
s1 := Sym(13)!(1,2)(3,4)(5,6)(7,8);
s2 := Sym(13)!(10,11);
s3 := Sym(13)!(12,13);
poly := sub<Sym(13)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```

to this polytope