Polytope of Type {6,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*1944o
if this polytope has a name.
Group : SmallGroup(1944,2341)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 54, 486, 162
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,18}*648i, {6,6}*648e
   9-fold quotients : {6,18}*216a, {6,18}*216b, {6,6}*216c, {6,6}*216d
   18-fold quotients : {6,9}*108, {3,6}*108
   27-fold quotients : {2,18}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
   54-fold quotients : {2,9}*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=<s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 2.
      81 facets:
         81 of {6}*12
      30 vertex figures:
         24 of {18}*36
         6 of {9}*18
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1> of order 3.
      72 facets:
         45 of {6}*12
         27 of {2}*4
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      27 facets:
         27 of {6}*12
      12 vertex figures:
         6 of {9}*18
         6 of {18}*36
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,18}

Twisty Puzzle