Polytope of Type {6,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*324d
if this polytope has a name.
Group : SmallGroup(324,40)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 18, 81, 27
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,9,2} of size 648
   {6,9,4} of size 1296
   {6,9,6} of size 1944
Vertex Figure Of :
   {2,6,9} of size 648
   {4,6,9} of size 1296
   {6,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,3}*108
   9-fold quotients : {6,3}*36
   27-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,18}*648e
   3-fold covers : {6,9}*972b, {18,9}*972d, {18,9}*972e, {18,9}*972g, {6,9}*972e
   4-fold covers : {6,36}*1296e, {12,18}*1296h, {6,9}*1296a, {12,9}*1296a
   5-fold covers : {6,45}*1620d
   6-fold covers : {6,18}*1944d, {18,18}*1944j, {18,18}*1944n, {18,18}*1944r, {6,18}*1944i, {6,18}*1944r
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
      9 facets:
         9 of {6}*12
      6 vertex figures:
         6 of {9}*18

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,9}

Twisty Puzzle