Polytope of Type {9,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*972b
if this polytope has a name.
Group : SmallGroup(972,102)
Rank : 3
Schlafli Type : {9,6}
Number of vertices, edges, etc : 81, 243, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {9,6,2} of size 1944
Vertex Figure Of :
   {2,9,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,6}*324b, {9,6}*324d
   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 : {18,6}*1944d
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {9}*18
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2> of order 3.
      36 facets:
         9 of {9}*18
         27 of {3}*6
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {9}*18
      33 vertex figures:
         24 of {6}*12
         9 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2, s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 9.
      12 facets:
         3 of {9}*18
         9 of {3}*6
      9 vertex figures:
         9 of {6}*12

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 := ( 4, 7)( 5, 8)( 6, 9)(10,20)(11,21)(12,19)(13,26)(14,27)(15,25)(16,23)(17,24)(18,22)(31,34)(32,35)(33,36)(37,47)(38,48)(39,46)(40,53)(41,54)(42,52)(43,50)(44,51)(45,49)(58,61)(59,62)(60,63)(64,74)(65,75)(66,73)(67,80)(68,81)(69,79)(70,77)(71,78)(72,76);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*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)(28,57)(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)!( 4, 7)( 5, 8)( 6, 9)(10,20)(11,21)(12,19)(13,26)(14,27)(15,25)(16,23)(17,24)(18,22)(31,34)(32,35)(33,36)(37,47)(38,48)(39,46)(40,53)(41,54)(42,52)(43,50)(44,51)(45,49)(58,61)(59,62)(60,63)(64,74)(65,75)(66,73)(67,80)(68,81)(69,79)(70,77)(71,78)(72,76);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1 >;
References : None.
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