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# Polytope of Type {4,9,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9,2}*144
if this polytope has a name.
Group : SmallGroup(144,109)
Rank : 4
Schlafli Type : {4,9,2}
Number of vertices, edges, etc : 4, 18, 9, 2
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,9,2,2} of size 288
{4,9,2,3} of size 432
{4,9,2,4} of size 576
{4,9,2,5} of size 720
{4,9,2,6} of size 864
{4,9,2,7} of size 1008
{4,9,2,8} of size 1152
{4,9,2,9} of size 1296
{4,9,2,10} of size 1440
{4,9,2,11} of size 1584
{4,9,2,12} of size 1728
{4,9,2,13} of size 1872
Vertex Figure Of :
{2,4,9,2} of size 288
{4,4,9,2} of size 1152
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,9,2}*288, {4,18,2}*288b, {4,18,2}*288c
3-fold covers : {4,27,2}*432, {4,9,6}*432
4-fold covers : {4,36,2}*576b, {4,36,2}*576c, {4,18,4}*576c, {8,9,2}*576, {4,18,2}*576, {4,9,4}*576a
5-fold covers : {4,45,2}*720
6-fold covers : {4,27,2}*864, {4,54,2}*864b, {4,54,2}*864c, {4,9,6}*864, {4,18,6}*864c, {4,18,6}*864d, {4,18,6}*864e, {12,9,2}*864, {12,18,2}*864c
7-fold covers : {4,63,2}*1008
8-fold covers : {4,36,4}*1152d, {4,36,4}*1152e, {4,18,2}*1152a, {8,9,2}*1152, {8,18,2}*1152a, {4,72,2}*1152c, {4,72,2}*1152d, {4,18,8}*1152b, {4,36,2}*1152b, {4,18,4}*1152b, {4,18,2}*1152b, {4,36,2}*1152c, {8,18,2}*1152b, {8,18,2}*1152c, {4,9,8}*1152, {4,9,4}*1152, {4,18,4}*1152c, {4,18,4}*1152f
9-fold covers : {4,81,2}*1296, {4,27,6}*1296, {4,9,18}*1296, {4,9,6}*1296c
10-fold covers : {4,18,10}*1440b, {20,18,2}*1440b, {4,45,2}*1440, {4,90,2}*1440b, {4,90,2}*1440c
11-fold covers : {4,99,2}*1584
12-fold covers : {4,108,2}*1728b, {4,108,2}*1728c, {4,54,4}*1728c, {8,27,2}*1728, {4,54,2}*1728, {4,27,4}*1728a, {4,36,6}*1728c, {4,36,6}*1728d, {4,36,6}*1728e, {4,36,6}*1728f, {4,18,12}*1728c, {24,9,2}*1728, {8,9,6}*1728, {4,18,12}*1728d, {4,9,6}*1728, {4,18,6}*1728a, {4,18,6}*1728b, {12,18,2}*1728a, {12,18,2}*1728b, {4,9,12}*1728
13-fold covers : {4,117,2}*1872
Permutation Representation (GAP) :
```s0 := ( 2, 7)( 3, 9)( 4,11)( 5,13)( 8,18)(10,20)(14,24)(21,30)(23,32)(25,33)
(27,34)(29,35);;
s1 := ( 1, 2)( 3, 6)( 4, 5)( 7,15)( 8,14)( 9,16)(10,12)(11,13)(17,23)(18,24)
(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);;
s2 := ( 1, 6)( 2, 4)( 3,14)( 5,10)( 7,11)( 8,23)( 9,24)(12,19)(13,20)(15,16)
(17,31)(18,32)(21,27)(22,28)(25,29)(26,36)(30,34)(33,35);;
s3 := (37,38);;
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,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(38)!( 2, 7)( 3, 9)( 4,11)( 5,13)( 8,18)(10,20)(14,24)(21,30)(23,32)
(25,33)(27,34)(29,35);
s1 := Sym(38)!( 1, 2)( 3, 6)( 4, 5)( 7,15)( 8,14)( 9,16)(10,12)(11,13)(17,23)
(18,24)(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);
s2 := Sym(38)!( 1, 6)( 2, 4)( 3,14)( 5,10)( 7,11)( 8,23)( 9,24)(12,19)(13,20)
(15,16)(17,31)(18,32)(21,27)(22,28)(25,29)(26,36)(30,34)(33,35);
s3 := Sym(38)!(37,38);
poly := sub<Sym(38)|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, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope