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}*1944s
if this polytope has a name.
Group : SmallGroup(1944,2345)
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}*648f, {6,6}*648g
   6-fold quotients : {6,18}*324b
   9-fold quotients : {6,6}*216b, {6,6}*216d
   18-fold quotients : {6,6}*108
   27-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   54-fold quotients : {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*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      81 facets:
         81 of {6}*12
      27 vertex figures:
         27 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*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=<s0*s1*s0*s1*s2*s1*s0*s1*s2*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*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=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      27 facets:
         27 of {6}*12
      12 vertex figures:
         6 of {18}*36
         6 of {9}*18
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      27 facets:
         27 of {6}*12
      12 vertex figures:
         6 of {18}*36
         6 of {9}*18
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 9.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,18}

Twisty Puzzle