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}*648i
if this polytope has a name.
Group : SmallGroup(648,554)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 18, 162, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,18,2} of size 1296
Vertex Figure Of :
   {2,6,18} of size 1296
   {3,6,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,18}*216a, {6,18}*216b, {6,6}*216d
   6-fold quotients : {6,9}*108
   9-fold quotients : {2,18}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
   18-fold quotients : {2,9}*36, {3,6}*36, {6,3}*36
   27-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {2,3}*12, {3,2}*12
   81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,36}*1296l, {12,18}*1296l
   3-fold covers : {18,18}*1944ad, {18,18}*1944ae, {6,18}*1944m, {6,18}*1944n, {6,18}*1944o, {6,54}*1944g
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.
      27 facets:
         27 of {6}*12
      12 vertex figures:
         6 of {18}*36
         6 of {9}*18
   P/N, where N=<s0*s1*s0*s1> of order 3.
      36 facets:
         27 of {2}*4
         9 of {6}*12
      6 vertex figures:
         6 of {18}*36
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36

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