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Polytope of Type {2,10,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,10,18}*720
if this polytope has a name.
Group : SmallGroup(720,402)
Rank : 4
Schlafli Type : {2,10,18}
Number of vertices, edges, etc : 2, 10, 90, 18
Order of s0s1s2s3 : 90
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,10,18,2} of size 1440
Vertex Figure Of :
   {2,2,10,18} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,10,6}*240
   5-fold quotients : {2,2,18}*144
   9-fold quotients : {2,10,2}*80
   10-fold quotients : {2,2,9}*72
   15-fold quotients : {2,2,6}*48
   18-fold quotients : {2,5,2}*40
   30-fold quotients : {2,2,3}*24
   45-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,10,36}*1440, {2,20,18}*1440a, {4,10,18}*1440
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 6,15)( 7,16)( 8,17)( 9,12)(10,13)(11,14)(21,30)(22,31)(23,32)(24,27)
(25,28)(26,29)(36,45)(37,46)(38,47)(39,42)(40,43)(41,44)(51,60)(52,61)(53,62)
(54,57)(55,58)(56,59)(66,75)(67,76)(68,77)(69,72)(70,73)(71,74)(81,90)(82,91)
(83,92)(84,87)(85,88)(86,89);;
s2 := ( 3, 6)( 4, 8)( 5, 7)( 9,15)(10,17)(11,16)(13,14)(18,37)(19,36)(20,38)
(21,34)(22,33)(23,35)(24,46)(25,45)(26,47)(27,43)(28,42)(29,44)(30,40)(31,39)
(32,41)(48,51)(49,53)(50,52)(54,60)(55,62)(56,61)(58,59)(63,82)(64,81)(65,83)
(66,79)(67,78)(68,80)(69,91)(70,90)(71,92)(72,88)(73,87)(74,89)(75,85)(76,84)
(77,86);;
s3 := ( 3,63)( 4,65)( 5,64)( 6,66)( 7,68)( 8,67)( 9,69)(10,71)(11,70)(12,72)
(13,74)(14,73)(15,75)(16,77)(17,76)(18,48)(19,50)(20,49)(21,51)(22,53)(23,52)
(24,54)(25,56)(26,55)(27,57)(28,59)(29,58)(30,60)(31,62)(32,61)(33,79)(34,78)
(35,80)(36,82)(37,81)(38,83)(39,85)(40,84)(41,86)(42,88)(43,87)(44,89)(45,91)
(46,90)(47,92);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(92)!(1,2);
s1 := Sym(92)!( 6,15)( 7,16)( 8,17)( 9,12)(10,13)(11,14)(21,30)(22,31)(23,32)
(24,27)(25,28)(26,29)(36,45)(37,46)(38,47)(39,42)(40,43)(41,44)(51,60)(52,61)
(53,62)(54,57)(55,58)(56,59)(66,75)(67,76)(68,77)(69,72)(70,73)(71,74)(81,90)
(82,91)(83,92)(84,87)(85,88)(86,89);
s2 := Sym(92)!( 3, 6)( 4, 8)( 5, 7)( 9,15)(10,17)(11,16)(13,14)(18,37)(19,36)
(20,38)(21,34)(22,33)(23,35)(24,46)(25,45)(26,47)(27,43)(28,42)(29,44)(30,40)
(31,39)(32,41)(48,51)(49,53)(50,52)(54,60)(55,62)(56,61)(58,59)(63,82)(64,81)
(65,83)(66,79)(67,78)(68,80)(69,91)(70,90)(71,92)(72,88)(73,87)(74,89)(75,85)
(76,84)(77,86);
s3 := Sym(92)!( 3,63)( 4,65)( 5,64)( 6,66)( 7,68)( 8,67)( 9,69)(10,71)(11,70)
(12,72)(13,74)(14,73)(15,75)(16,77)(17,76)(18,48)(19,50)(20,49)(21,51)(22,53)
(23,52)(24,54)(25,56)(26,55)(27,57)(28,59)(29,58)(30,60)(31,62)(32,61)(33,79)
(34,78)(35,80)(36,82)(37,81)(38,83)(39,85)(40,84)(41,86)(42,88)(43,87)(44,89)
(45,91)(46,90)(47,92);
poly := sub<Sym(92)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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