Polytope of Type {6,15}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,15}*540
if this polytope has a name.
Group : SmallGroup(540,54)
Rank : 3
Schlafli Type : {6,15}
Number of vertices, edges, etc : 18, 135, 45
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,15,2} of size 1080
Vertex Figure Of :
   {2,6,15} of size 1080
   {3,6,15} of size 1620
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,15}*180
   5-fold quotients : {6,3}*108
   9-fold quotients : {2,15}*60
   15-fold quotients : {6,3}*36
   27-fold quotients : {2,5}*20
   45-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,30}*1080b
   3-fold covers : {6,45}*1620a, {6,45}*1620b, {6,45}*1620c, {6,45}*1620d, {6,15}*1620, {18,15}*1620
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1> of order 3.
      25 facets:
         15 of {2}*4
         10 of {6}*12
      6 vertex figures:
         6 of {15}*30

Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45);;
s1 := ( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)(16,33)(17,31)(18,32)(19,45)(20,43)(21,44)(22,42)(23,40)(24,41)(25,39)(26,37)(27,38)(28,36)(29,34)(30,35);;
s2 := ( 1,19)( 2,20)( 3,21)( 4,16)( 5,17)( 6,18)( 7,28)( 8,29)( 9,30)(10,25)(11,26)(12,27)(13,22)(14,23)(15,24)(31,34)(32,35)(33,36)(37,43)(38,44)(39,45);;
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, 
s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0, 
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(45)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45);
s1 := Sym(45)!( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)(16,33)(17,31)(18,32)(19,45)(20,43)(21,44)(22,42)(23,40)(24,41)(25,39)(26,37)(27,38)(28,36)(29,34)(30,35);
s2 := Sym(45)!( 1,19)( 2,20)( 3,21)( 4,16)( 5,17)( 6,18)( 7,28)( 8,29)( 9,30)(10,25)(11,26)(12,27)(13,22)(14,23)(15,24)(31,34)(32,35)(33,36)(37,43)(38,44)(39,45);
poly := sub<Sym(45)|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, 
s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0, 
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 {6,15}

Twisty Puzzle