Polytope of Type {10,18}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope