Polytope of Type {4,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18}*648
if this polytope has a name.
Group : SmallGroup(648,252)
Rank : 3
Schlafli Type : {4,18}
Number of vertices, edges, etc : 18, 162, 81
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {4,18,2} of size 1296
Vertex Figure Of :
   {2,4,18} of size 1296
Quotients (Maximal Quotients in Boldface) :
   9-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,18}*1296a
   3-fold covers : {4,18}*1944a, {12,18}*1944a, {12,18}*1944b, {12,18}*1944g
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      27 facets:
         27 of {4}*8
      6 vertex figures:
         6 of {18}*36
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      27 facets:
         27 of {4}*8
      12 vertex figures:
         9 of {6}*12
         3 of {18}*36

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {4,18}

Twisty Puzzle