Polytope of Type {9,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,18}*972c
if this polytope has a name.
Group : SmallGroup(972,103)
Rank : 3
Schlafli Type : {9,18}
Number of vertices, edges, etc : 27, 243, 54
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {9,18,2} of size 1944
Vertex Figure Of :
   {2,9,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,6}*324c, {3,18}*324
   9-fold quotients : {3,6}*108
   27-fold quotients : {3,6}*36
   81-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,18}*1944h
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope