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# Polytope of Type {4,9}

Atlas Canonical Name : {4,9}*72
if this polytope has a name.
Group : SmallGroup(72,15)
Rank : 3
Schlafli Type : {4,9}
Number of vertices, edges, etc : 4, 18, 9
Order of s0s1s2 : 9
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,9,2} of size 144
{4,9,4} of size 288
{4,9,6} of size 432
{4,9,4} of size 576
{4,9,8} of size 1152
{4,9,18} of size 1296
{4,9,6} of size 1296
{4,9,6} of size 1296
{4,9,6} of size 1296
{4,9,6} of size 1296
{4,9,6} of size 1728
{4,9,12} of size 1728
Vertex Figure Of :
{2,4,9} of size 144
{4,4,9} of size 576
{4,4,9} of size 1152
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,9}*144, {4,18}*144b, {4,18}*144c
3-fold covers : {4,27}*216
4-fold covers : {4,36}*288b, {4,36}*288c, {8,9}*288, {4,18}*288
5-fold covers : {4,45}*360
6-fold covers : {4,27}*432, {4,54}*432b, {4,54}*432c, {12,9}*432, {12,18}*432c
7-fold covers : {4,63}*504
8-fold covers : {4,18}*576a, {8,9}*576, {8,18}*576a, {4,72}*576c, {4,72}*576d, {4,36}*576b, {4,18}*576b, {4,36}*576c, {8,18}*576b, {8,18}*576c
9-fold covers : {4,81}*648
10-fold covers : {20,18}*720b, {4,45}*720, {4,90}*720b, {4,90}*720c
11-fold covers : {4,99}*792
12-fold covers : {4,108}*864b, {4,108}*864c, {8,27}*864, {4,54}*864, {24,9}*864, {12,18}*864a, {12,18}*864b
13-fold covers : {4,117}*936
14-fold covers : {28,18}*1008b, {4,63}*1008, {4,126}*1008b, {4,126}*1008c
15-fold covers : {4,135}*1080
16-fold covers : {4,36}*1152b, {4,36}*1152c, {8,9}*1152, {8,18}*1152a, {8,36}*1152c, {8,36}*1152d, {8,18}*1152b, {8,18}*1152c, {4,144}*1152c, {4,144}*1152d, {4,36}*1152d, {8,36}*1152e, {8,36}*1152f, {4,18}*1152a, {8,18}*1152d, {8,18}*1152e, {8,18}*1152f, {8,36}*1152g, {8,36}*1152h, {4,72}*1152c, {4,72}*1152d, {8,18}*1152g, {4,36}*1152e, {4,72}*1152e, {4,18}*1152b, {4,72}*1152f
17-fold covers : {4,153}*1224
18-fold covers : {4,81}*1296, {4,162}*1296b, {4,162}*1296c, {12,27}*1296, {12,54}*1296c, {36,9}*1296, {36,18}*1296d, {12,9}*1296c, {12,18}*1296k
19-fold covers : {4,171}*1368
20-fold covers : {4,180}*1440b, {4,180}*1440c, {8,45}*1440, {20,18}*1440, {4,90}*1440
21-fold covers : {4,189}*1512
22-fold covers : {44,18}*1584b, {4,99}*1584, {4,198}*1584b, {4,198}*1584c
23-fold covers : {4,207}*1656
24-fold covers : {4,54}*1728a, {8,27}*1728, {8,54}*1728a, {4,216}*1728c, {4,216}*1728d, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c, {24,9}*1728, {24,18}*1728a, {12,36}*1728e, {12,36}*1728f, {12,18}*1728c, {12,36}*1728g, {24,18}*1728b, {24,18}*1728c, {24,18}*1728d, {24,18}*1728e, {12,18}*1728d, {12,36}*1728h, {12,9}*1728, {12,36}*1728i
25-fold covers : {4,225}*1800
26-fold covers : {52,18}*1872b, {4,117}*1872, {4,234}*1872b, {4,234}*1872c
27-fold covers : {4,243}*1944, {4,9}*1944, {12,9}*1944a, {12,9}*1944b, {12,9}*1944c
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);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, 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(36)!( 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(36)!( 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(36)!( 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);
poly := sub<Sym(36)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, 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 >;

```
References : None.
to this polytope