Polytope of Type {18,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,9}*972c
if this polytope has a name.
Group : SmallGroup(972,103)
Rank : 3
Schlafli Type : {18,9}
Number of vertices, edges, etc : 54, 243, 27
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 :
   {18,9,2} of size 1944
Vertex Figure Of :
   {2,18,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,9}*324c, {18,3}*324
   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 : {18,18}*1944g
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.
      9 facets:
         9 of {18}*36
      18 vertex figures:
         18 of {9}*18

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {18,9}

Twisty Puzzle