Polytope of Type {18,9}

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

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

References : None.
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