Polytope of Type {18,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*1944n
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 486, 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 :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,6}*648b, {18,6}*648i, {6,6}*648g
   6-fold quotients : {18,6}*324a
   9-fold quotients : {18,6}*216a, {18,6}*216b, {6,6}*216b, {6,6}*216d
   18-fold quotients : {9,6}*108, {6,6}*108
   27-fold quotients : {18,2}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
   54-fold quotients : {9,2}*36, {3,6}*36, {6,3}*36
   81-fold quotients : {2,6}*24, {6,2}*24
   162-fold quotients : {2,3}*12, {3,2}*12
   243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      30 facets:
         6 of {9}*18
         24 of {18}*36
      81 vertex figures:
         81 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1> of order 2.
      27 facets:
         27 of {18}*36
      81 vertex figures:
         81 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {18}*36
      72 vertex figures:
         45 of {6}*12
         27 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         6 of {9}*18
         6 of {18}*36
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 9.
      6 facets:
         6 of {18}*36
      30 vertex figures:
         12 of {6}*12
         18 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 9.
      6 facets:
         6 of {18}*36
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1> of order 9.
      6 facets:
         6 of {18}*36
      18 vertex figures:
         18 of {6}*12

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {18,6}

Twisty Puzzle