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# Polytope of Type {2,9,4}

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

```

to this polytope