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# Polytope of Type {2,90}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,90}*360
if this polytope has a name.
Group : SmallGroup(360,49)
Rank : 3
Schlafli Type : {2,90}
Number of vertices, edges, etc : 2, 90, 90
Order of s0s1s2 : 90
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,90,2} of size 720
{2,90,4} of size 1440
{2,90,4} of size 1440
{2,90,4} of size 1440
Vertex Figure Of :
{2,2,90} of size 720
{3,2,90} of size 1080
{4,2,90} of size 1440
{5,2,90} of size 1800
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,45}*180
3-fold quotients : {2,30}*120
5-fold quotients : {2,18}*72
6-fold quotients : {2,15}*60
9-fold quotients : {2,10}*40
10-fold quotients : {2,9}*36
15-fold quotients : {2,6}*24
18-fold quotients : {2,5}*20
30-fold quotients : {2,3}*12
45-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,180}*720, {4,90}*720a
3-fold covers : {2,270}*1080, {6,90}*1080a, {6,90}*1080b
4-fold covers : {4,180}*1440a, {2,360}*1440, {8,90}*1440, {4,90}*1440
5-fold covers : {2,450}*1800, {10,90}*1800b, {10,90}*1800c
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 6,15)( 7,17)( 8,16)( 9,12)(10,14)(11,13)(18,34)(19,33)(20,35)
(21,46)(22,45)(23,47)(24,43)(25,42)(26,44)(27,40)(28,39)(29,41)(30,37)(31,36)
(32,38)(49,50)(51,60)(52,62)(53,61)(54,57)(55,59)(56,58)(63,79)(64,78)(65,80)
(66,91)(67,90)(68,92)(69,88)(70,87)(71,89)(72,85)(73,84)(74,86)(75,82)(76,81)
(77,83);;
s2 := ( 3,66)( 4,68)( 5,67)( 6,63)( 7,65)( 8,64)( 9,75)(10,77)(11,76)(12,72)
(13,74)(14,73)(15,69)(16,71)(17,70)(18,51)(19,53)(20,52)(21,48)(22,50)(23,49)
(24,60)(25,62)(26,61)(27,57)(28,59)(29,58)(30,54)(31,56)(32,55)(33,82)(34,81)
(35,83)(36,79)(37,78)(38,80)(39,91)(40,90)(41,92)(42,88)(43,87)(44,89)(45,85)
(46,84)(47,86);;
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*s1*s0*s1, s0*s2*s0*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*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*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(92)!(1,2);
s1 := Sym(92)!( 4, 5)( 6,15)( 7,17)( 8,16)( 9,12)(10,14)(11,13)(18,34)(19,33)
(20,35)(21,46)(22,45)(23,47)(24,43)(25,42)(26,44)(27,40)(28,39)(29,41)(30,37)
(31,36)(32,38)(49,50)(51,60)(52,62)(53,61)(54,57)(55,59)(56,58)(63,79)(64,78)
(65,80)(66,91)(67,90)(68,92)(69,88)(70,87)(71,89)(72,85)(73,84)(74,86)(75,82)
(76,81)(77,83);
s2 := Sym(92)!( 3,66)( 4,68)( 5,67)( 6,63)( 7,65)( 8,64)( 9,75)(10,77)(11,76)
(12,72)(13,74)(14,73)(15,69)(16,71)(17,70)(18,51)(19,53)(20,52)(21,48)(22,50)
(23,49)(24,60)(25,62)(26,61)(27,57)(28,59)(29,58)(30,54)(31,56)(32,55)(33,82)
(34,81)(35,83)(36,79)(37,78)(38,80)(39,91)(40,90)(41,92)(42,88)(43,87)(44,89)
(45,85)(46,84)(47,86);
poly := sub<Sym(92)|s0,s1,s2>;

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

```

to this polytope