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Polytope of Type {3,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,18}*324
if this polytope has a name.
Group : SmallGroup(324,41)
Rank : 3
Schlafli Type : {3,18}
Number of vertices, edges, etc : 9, 81, 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 :
   {3,18,2} of size 648
   {3,18,4} of size 1296
   {3,18,6} of size 1944
Vertex Figure Of :
   {2,3,18} of size 648
   {4,3,18} of size 1296
   {6,3,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*108
   9-fold quotients : {3,6}*36
   27-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,18}*648h
   3-fold covers : {3,18}*972a, {9,18}*972c, {9,18}*972d, {9,18}*972i, {9,18}*972j, {3,18}*972b
   4-fold covers : {12,18}*1296d, {6,36}*1296h, {3,36}*1296, {3,18}*1296a
   5-fold covers : {15,18}*1620
   6-fold covers : {6,18}*1944b, {18,18}*1944h, {18,18}*1944l, {18,18}*1944y, {18,18}*1944ab, {6,18}*1944k, {6,18}*1944u
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 6)( 7, 8)(10,19)(11,21)(12,20)(13,24)(14,23)(15,22)(16,26)
(17,25)(18,27)(28,57)(29,56)(30,55)(31,59)(32,58)(33,60)(34,61)(35,63)(36,62)
(37,75)(38,74)(39,73)(40,77)(41,76)(42,78)(43,79)(44,81)(45,80)(46,66)(47,65)
(48,64)(49,68)(50,67)(51,69)(52,70)(53,72)(54,71);;
s1 := ( 1,43)( 2,45)( 3,44)( 4,37)( 5,39)( 6,38)( 7,40)( 8,42)( 9,41)(10,31)
(11,33)(12,32)(13,34)(14,36)(15,35)(16,28)(17,30)(18,29)(19,47)(20,46)(21,48)
(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(55,72)(56,71)(57,70)(58,66)(59,65)
(60,64)(61,69)(62,68)(63,67)(74,75)(77,78)(80,81);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)
(23,27)(24,26)(28,57)(29,56)(30,55)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)
(37,66)(38,65)(39,64)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,75)(47,74)
(48,73)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76);;
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*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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, 6)( 7, 8)(10,19)(11,21)(12,20)(13,24)(14,23)(15,22)
(16,26)(17,25)(18,27)(28,57)(29,56)(30,55)(31,59)(32,58)(33,60)(34,61)(35,63)
(36,62)(37,75)(38,74)(39,73)(40,77)(41,76)(42,78)(43,79)(44,81)(45,80)(46,66)
(47,65)(48,64)(49,68)(50,67)(51,69)(52,70)(53,72)(54,71);
s1 := Sym(81)!( 1,43)( 2,45)( 3,44)( 4,37)( 5,39)( 6,38)( 7,40)( 8,42)( 9,41)
(10,31)(11,33)(12,32)(13,34)(14,36)(15,35)(16,28)(17,30)(18,29)(19,47)(20,46)
(21,48)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(55,72)(56,71)(57,70)(58,66)
(59,65)(60,64)(61,69)(62,68)(63,67)(74,75)(77,78)(80,81);
s2 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)
(22,25)(23,27)(24,26)(28,57)(29,56)(30,55)(31,63)(32,62)(33,61)(34,60)(35,59)
(36,58)(37,66)(38,65)(39,64)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,75)
(47,74)(48,73)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76);
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*s0*s1*s0*s1, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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.
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