Polytope of Type {9,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,18}*972b
if this polytope has a name.
Group : SmallGroup(972,102)
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
   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}*1944f
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)(28,57)(29,56)(30,55)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)
(37,71)(38,70)(39,72)(40,68)(41,67)(42,69)(43,65)(44,64)(45,66)(46,76)(47,78)
(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);;
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)(55,61)(56,63)(57,62)(59,60)(64,71)
(65,70)(66,72)(67,68)(73,81)(74,80)(75,79)(76,78);;
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)(28,57)(29,56)(30,55)(31,60)(32,59)(33,58)(34,63)(35,62)(36,61)
(37,78)(38,77)(39,76)(40,81)(41,80)(42,79)(43,75)(44,74)(45,73)(46,72)(47,71)
(48,70)(49,66)(50,65)(51,64)(52,69)(53,68)(54,67);;
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, s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*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*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*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)(28,57)(29,56)(30,55)(31,63)(32,62)(33,61)(34,60)(35,59)
(36,58)(37,71)(38,70)(39,72)(40,68)(41,67)(42,69)(43,65)(44,64)(45,66)(46,76)
(47,78)(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);
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)(55,61)(56,63)(57,62)(59,60)
(64,71)(65,70)(66,72)(67,68)(73,81)(74,80)(75,79)(76,78);
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)(28,57)(29,56)(30,55)(31,60)(32,59)(33,58)(34,63)(35,62)
(36,61)(37,78)(38,77)(39,76)(40,81)(41,80)(42,79)(43,75)(44,74)(45,73)(46,72)
(47,71)(48,70)(49,66)(50,65)(51,64)(52,69)(53,68)(54,67);
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, s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*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*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 >;

References : None.
to this polytope