Polytope of Type {6,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*972b
if this polytope has a name.
Group : SmallGroup(972,102)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 54, 243, 81
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 :
   {6,9,2} of size 1944
Vertex Figure Of :
   {2,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,9}*324b, {6,9}*324d
   9-fold quotients : {6,3}*108
   27-fold quotients : {6,3}*36
   81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,18}*1944d
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
      27 facets:
         27 of {6}*12
      18 vertex figures:
         18 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2> of order 3.
      27 facets:
         27 of {6}*12
      36 vertex figures:
         9 of {9}*18
         27 of {3}*6
   P/N, where N=<s0*s1*s0*s1> of order 3.
      33 facets:
         9 of {2}*4
         24 of {6}*12
      18 vertex figures:
         18 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 9.
      9 facets:
         9 of {6}*12
      12 vertex figures:
         3 of {9}*18
         9 of {3}*6

Permutation Representation (GAP) : 
s0 := ( 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);;
s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,16)(11,18)(12,17)(14,15)(19,22)(20,24)(21,23)(26,27)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,81)(35,80)(36,79)(37,57)(38,56)(39,55)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,39)(11,38)(12,37)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,47)(20,46)(21,48)(22,53)(23,52)(24,54)(25,50)(26,49)(27,51)(55,57)(58,63)(59,62)(60,61)(64,65)(67,71)(68,70)(69,72)(74,75)(76,79)(77,81)(78,80);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := 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);
s1 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,16)(11,18)(12,17)(14,15)(19,22)(20,24)(21,23)(26,27)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,81)(35,80)(36,79)(37,57)(38,56)(39,55)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64);
s2 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,39)(11,38)(12,37)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,47)(20,46)(21,48)(22,53)(23,52)(24,54)(25,50)(26,49)(27,51)(55,57)(58,63)(59,62)(60,61)(64,65)(67,71)(68,70)(69,72)(74,75)(76,79)(77,81)(78,80);
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0 >;
References : None.
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