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# Polytope of Type {2,4,9}

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

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

```

to this polytope