Polytope of Type {18,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,10}*360
Also Known As : {18,10|2}. if this polytope has another name.
Group : SmallGroup(360,45)
Rank : 3
Schlafli Type : {18,10}
Number of vertices, edges, etc : 18, 90, 10
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,10,2} of size 720
   {18,10,4} of size 1440
   {18,10,5} of size 1800
Vertex Figure Of :
   {2,18,10} of size 720
   {4,18,10} of size 1440
   {4,18,10} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,10}*120
   5-fold quotients : {18,2}*72
   9-fold quotients : {2,10}*40
   10-fold quotients : {9,2}*36
   15-fold quotients : {6,2}*24
   18-fold quotients : {2,5}*20
   30-fold quotients : {3,2}*12
   45-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,10}*720, {18,20}*720a
   3-fold covers : {54,10}*1080, {18,30}*1080a, {18,30}*1080b
   4-fold covers : {72,10}*1440, {18,40}*1440, {36,20}*1440, {18,20}*1440
   5-fold covers : {18,50}*1800, {90,10}*1800a, {90,10}*1800b
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(16,32)(17,31)(18,33)(19,35)(20,34)
(21,36)(22,38)(23,37)(24,39)(25,41)(26,40)(27,42)(28,44)(29,43)(30,45)(47,48)
(50,51)(53,54)(56,57)(59,60)(61,77)(62,76)(63,78)(64,80)(65,79)(66,81)(67,83)
(68,82)(69,84)(70,86)(71,85)(72,87)(73,89)(74,88)(75,90);;
s1 := ( 1,16)( 2,18)( 3,17)( 4,28)( 5,30)( 6,29)( 7,25)( 8,27)( 9,26)(10,22)
(11,24)(12,23)(13,19)(14,21)(15,20)(31,32)(34,44)(35,43)(36,45)(37,41)(38,40)
(39,42)(46,61)(47,63)(48,62)(49,73)(50,75)(51,74)(52,70)(53,72)(54,71)(55,67)
(56,69)(57,68)(58,64)(59,66)(60,65)(76,77)(79,89)(80,88)(81,90)(82,86)(83,85)
(84,87);;
s2 := ( 1,49)( 2,50)( 3,51)( 4,46)( 5,47)( 6,48)( 7,58)( 8,59)( 9,60)(10,55)
(11,56)(12,57)(13,52)(14,53)(15,54)(16,64)(17,65)(18,66)(19,61)(20,62)(21,63)
(22,73)(23,74)(24,75)(25,70)(26,71)(27,72)(28,67)(29,68)(30,69)(31,79)(32,80)
(33,81)(34,76)(35,77)(36,78)(37,88)(38,89)(39,90)(40,85)(41,86)(42,87)(43,82)
(44,83)(45,84);;
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*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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(90)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(16,32)(17,31)(18,33)(19,35)
(20,34)(21,36)(22,38)(23,37)(24,39)(25,41)(26,40)(27,42)(28,44)(29,43)(30,45)
(47,48)(50,51)(53,54)(56,57)(59,60)(61,77)(62,76)(63,78)(64,80)(65,79)(66,81)
(67,83)(68,82)(69,84)(70,86)(71,85)(72,87)(73,89)(74,88)(75,90);
s1 := Sym(90)!( 1,16)( 2,18)( 3,17)( 4,28)( 5,30)( 6,29)( 7,25)( 8,27)( 9,26)
(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)(31,32)(34,44)(35,43)(36,45)(37,41)
(38,40)(39,42)(46,61)(47,63)(48,62)(49,73)(50,75)(51,74)(52,70)(53,72)(54,71)
(55,67)(56,69)(57,68)(58,64)(59,66)(60,65)(76,77)(79,89)(80,88)(81,90)(82,86)
(83,85)(84,87);
s2 := Sym(90)!( 1,49)( 2,50)( 3,51)( 4,46)( 5,47)( 6,48)( 7,58)( 8,59)( 9,60)
(10,55)(11,56)(12,57)(13,52)(14,53)(15,54)(16,64)(17,65)(18,66)(19,61)(20,62)
(21,63)(22,73)(23,74)(24,75)(25,70)(26,71)(27,72)(28,67)(29,68)(30,69)(31,79)
(32,80)(33,81)(34,76)(35,77)(36,78)(37,88)(38,89)(39,90)(40,85)(41,86)(42,87)
(43,82)(44,83)(45,84);
poly := sub<Sym(90)|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*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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