Polytope of Type {9,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,18}*324
if this polytope has a name.
Group : SmallGroup(324,36)
Rank : 3
Schlafli Type : {9,18}
Number of vertices, edges, etc : 9, 81, 18
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {9,18,2} of size 648
   {9,18,4} of size 1296
   {9,18,6} of size 1944
Vertex Figure Of :
   {2,9,18} of size 648
   {4,9,18} of size 1296
   {6,9,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,6}*108
   9-fold quotients : {9,2}*36, {3,6}*36
   27-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,18}*648c
   3-fold covers : {9,18}*972a, {27,18}*972
   4-fold covers : {36,18}*1296b, {18,36}*1296c, {9,18}*1296a, {9,36}*1296
   5-fold covers : {45,18}*1620
   6-fold covers : {18,18}*1944a, {54,18}*1944b, {18,18}*1944af
Permutation Representation (GAP) :

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