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# Polytope of Type {4,36}

Atlas Canonical Name : {4,36}*288a
Also Known As : {4,36|2}. if this polytope has another name.
Group : SmallGroup(288,92)
Rank : 3
Schlafli Type : {4,36}
Number of vertices, edges, etc : 4, 72, 36
Order of s0s1s2 : 36
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,36,2} of size 576
{4,36,4} of size 1152
{4,36,4} of size 1152
{4,36,4} of size 1152
{4,36,6} of size 1728
{4,36,6} of size 1728
Vertex Figure Of :
{2,4,36} of size 576
{4,4,36} of size 1152
{6,4,36} of size 1728
{3,4,36} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,36}*144, {4,18}*144a
3-fold quotients : {4,12}*96a
4-fold quotients : {2,18}*72
6-fold quotients : {2,12}*48, {4,6}*48a
8-fold quotients : {2,9}*36
9-fold quotients : {4,4}*32
12-fold quotients : {2,6}*24
18-fold quotients : {2,4}*16, {4,2}*16
24-fold quotients : {2,3}*12
36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,72}*576a, {4,36}*576a, {4,72}*576b, {8,36}*576a, {8,36}*576b
3-fold covers : {4,108}*864a, {12,36}*864a, {12,36}*864b
4-fold covers : {8,36}*1152a, {4,72}*1152a, {8,72}*1152a, {8,72}*1152b, {8,72}*1152c, {8,72}*1152d, {16,36}*1152a, {4,144}*1152a, {16,36}*1152b, {4,144}*1152b, {4,36}*1152a, {4,72}*1152b, {8,36}*1152b, {4,36}*1152d
5-fold covers : {20,36}*1440, {4,180}*1440a
6-fold covers : {4,216}*1728a, {4,108}*1728a, {4,216}*1728b, {8,108}*1728a, {8,108}*1728b, {24,36}*1728a, {12,36}*1728a, {12,36}*1728b, {24,36}*1728b, {12,72}*1728a, {12,72}*1728b, {24,36}*1728c, {12,72}*1728c, {12,72}*1728d, {24,36}*1728d
Permutation Representation (GAP) :
```s0 := (37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)
(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72);;
s1 := ( 1,37)( 2,39)( 3,38)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,46)
(11,48)(12,47)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,55)(20,57)(21,56)
(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(28,64)(29,66)(30,65)(31,71)(32,70)
(33,72)(34,68)(35,67)(36,69);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,22)(20,24)
(21,23)(25,26)(28,31)(29,33)(30,32)(34,35)(37,58)(38,60)(39,59)(40,55)(41,57)
(42,56)(43,62)(44,61)(45,63)(46,67)(47,69)(48,68)(49,64)(50,66)(51,65)(52,71)
(53,70)(54,72);;
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*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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(72)!(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)
(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72);
s1 := Sym(72)!( 1,37)( 2,39)( 3,38)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)
(10,46)(11,48)(12,47)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,55)(20,57)
(21,56)(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(28,64)(29,66)(30,65)(31,71)
(32,70)(33,72)(34,68)(35,67)(36,69);
s2 := Sym(72)!( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,22)
(20,24)(21,23)(25,26)(28,31)(29,33)(30,32)(34,35)(37,58)(38,60)(39,59)(40,55)
(41,57)(42,56)(43,62)(44,61)(45,63)(46,67)(47,69)(48,68)(49,64)(50,66)(51,65)
(52,71)(53,70)(54,72);
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, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope