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# Polytope of Type {9,4}

Atlas Canonical Name : {9,4}*144
if this polytope has a name.
Group : SmallGroup(144,109)
Rank : 3
Schlafli Type : {9,4}
Number of vertices, edges, etc : 18, 36, 8
Order of s0s1s2 : 18
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{9,4,2} of size 288
{9,4,4} of size 576
{9,4,6} of size 864
{9,4,4} of size 1152
{9,4,8} of size 1152
{9,4,10} of size 1440
{9,4,12} of size 1728
{9,4,3} of size 1728
Vertex Figure Of :
{2,9,4} of size 288
{4,9,4} of size 576
{6,9,4} of size 864
{4,9,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9,4}*72
3-fold quotients : {3,4}*48
4-fold quotients : {9,2}*36
6-fold quotients : {3,4}*24
12-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {9,8}*288, {18,4}*288
3-fold covers : {27,4}*432, {9,12}*432
4-fold covers : {9,8}*576, {36,4}*576b, {18,4}*576b, {36,4}*576c, {18,8}*576b, {18,8}*576c
5-fold covers : {45,4}*720
6-fold covers : {27,8}*864, {54,4}*864, {9,24}*864, {18,12}*864a, {18,12}*864b
7-fold covers : {63,4}*1008
8-fold covers : {9,8}*1152, {18,8}*1152a, {36,4}*1152d, {36,8}*1152e, {36,8}*1152f, {18,4}*1152a, {18,8}*1152d, {18,8}*1152e, {18,8}*1152f, {36,8}*1152g, {36,8}*1152h, {72,4}*1152c, {72,4}*1152d, {18,8}*1152g, {36,4}*1152e, {72,4}*1152e, {18,4}*1152b, {72,4}*1152f
9-fold covers : {81,4}*1296, {27,12}*1296, {9,36}*1296, {9,12}*1296c
10-fold covers : {45,8}*1440, {18,20}*1440, {90,4}*1440
11-fold covers : {99,4}*1584
12-fold covers : {27,8}*1728, {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c, {9,24}*1728, {36,12}*1728e, {36,12}*1728f, {18,12}*1728c, {36,12}*1728g, {18,24}*1728b, {18,24}*1728c, {18,24}*1728d, {18,24}*1728e, {18,12}*1728d, {36,12}*1728h, {9,12}*1728
13-fold covers : {117,4}*1872
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(13,29)(14,31)(15,30)(16,32)(17,25)
(18,27)(19,26)(20,28)(21,33)(22,35)(23,34)(24,36)(38,39)(41,45)(42,47)(43,46)
(44,48)(49,65)(50,67)(51,66)(52,68)(53,61)(54,63)(55,62)(56,64)(57,69)(58,71)
(59,70)(60,72);;
s1 := ( 1,13)( 2,14)( 3,16)( 4,15)( 5,21)( 6,22)( 7,24)( 8,23)( 9,17)(10,18)
(11,20)(12,19)(25,29)(26,30)(27,32)(28,31)(35,36)(37,49)(38,50)(39,52)(40,51)
(41,57)(42,58)(43,60)(44,59)(45,53)(46,54)(47,56)(48,55)(61,65)(62,66)(63,68)
(64,67)(71,72);;
s2 := ( 1,40)( 2,39)( 3,38)( 4,37)( 5,44)( 6,43)( 7,42)( 8,41)( 9,48)(10,47)
(11,46)(12,45)(13,52)(14,51)(15,50)(16,49)(17,56)(18,55)(19,54)(20,53)(21,60)
(22,59)(23,58)(24,57)(25,64)(26,63)(27,62)(28,61)(29,68)(30,67)(31,66)(32,65)
(33,72)(34,71)(35,70)(36,69);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
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(72)!( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(13,29)(14,31)(15,30)(16,32)
(17,25)(18,27)(19,26)(20,28)(21,33)(22,35)(23,34)(24,36)(38,39)(41,45)(42,47)
(43,46)(44,48)(49,65)(50,67)(51,66)(52,68)(53,61)(54,63)(55,62)(56,64)(57,69)
(58,71)(59,70)(60,72);
s1 := Sym(72)!( 1,13)( 2,14)( 3,16)( 4,15)( 5,21)( 6,22)( 7,24)( 8,23)( 9,17)
(10,18)(11,20)(12,19)(25,29)(26,30)(27,32)(28,31)(35,36)(37,49)(38,50)(39,52)
(40,51)(41,57)(42,58)(43,60)(44,59)(45,53)(46,54)(47,56)(48,55)(61,65)(62,66)
(63,68)(64,67)(71,72);
s2 := Sym(72)!( 1,40)( 2,39)( 3,38)( 4,37)( 5,44)( 6,43)( 7,42)( 8,41)( 9,48)
(10,47)(11,46)(12,45)(13,52)(14,51)(15,50)(16,49)(17,56)(18,55)(19,54)(20,53)
(21,60)(22,59)(23,58)(24,57)(25,64)(26,63)(27,62)(28,61)(29,68)(30,67)(31,66)
(32,65)(33,72)(34,71)(35,70)(36,69);
poly := sub<Sym(72)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope