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}*1944m
if this polytope has a name.
Group : SmallGroup(1944,2340)
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}*648a, {6,18}*648i, {6,6}*648f
   6-fold quotients : {6,9}*324a
   9-fold quotients : {6,18}*216a, {6,18}*216b, {6,6}*216a, {6,6}*216d
   18-fold quotients : {6,9}*108, {6,3}*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*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      81 facets:
         81 of {6}*12
      36 vertex figures:
         18 of {18}*36
         18 of {9}*18
   P/N, where N=<s0*s2*s1*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*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*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*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*s2*s1*s2*s1*s0*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*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*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*s1*s0*s1> of order 3.
      72 facets:
         27 of {2}*4
         45 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*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1> of order 6.
      27 facets:
         27 of {6}*12
      12 vertex figures:
         6 of {18}*36
         6 of {9}*18
   P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 9.
      30 facets:
         12 of {6}*12
         18 of {2}*4
      6 vertex figures:
         6 of {18}*36
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 9.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*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 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,64)(38,66)(39,65)(40,67)(41,69)(42,68)(43,70)(44,72)(45,71)(46,73)(47,75)(48,74)(49,76)(50,78)(51,77)(52,79)(53,81)(54,80);;
s1 := ( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,51)(11,49)(12,50)(13,48)(14,46)(15,47)(16,54)(17,52)(18,53)(19,41)(20,42)(21,40)(22,38)(23,39)(24,37)(25,44)(26,45)(27,43)(58,61)(59,62)(60,63)(64,78)(65,76)(66,77)(67,75)(68,73)(69,74)(70,81)(71,79)(72,80);;
s2 := ( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(19,22)(20,23)(21,24)(28,37)(29,38)(30,39)(31,43)(32,44)(33,45)(34,40)(35,41)(36,42)(46,49)(47,50)(48,51)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69)(73,76)(74,77)(75,78);;
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*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*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)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,64)(38,66)(39,65)(40,67)(41,69)(42,68)(43,70)(44,72)(45,71)(46,73)(47,75)(48,74)(49,76)(50,78)(51,77)(52,79)(53,81)(54,80);
s1 := Sym(81)!( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,51)(11,49)(12,50)(13,48)(14,46)(15,47)(16,54)(17,52)(18,53)(19,41)(20,42)(21,40)(22,38)(23,39)(24,37)(25,44)(26,45)(27,43)(58,61)(59,62)(60,63)(64,78)(65,76)(66,77)(67,75)(68,73)(69,74)(70,81)(71,79)(72,80);
s2 := Sym(81)!( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(19,22)(20,23)(21,24)(28,37)(29,38)(30,39)(31,43)(32,44)(33,45)(34,40)(35,41)(36,42)(46,49)(47,50)(48,51)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69)(73,76)(74,77)(75,78);
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*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*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