Polytope of Type {6,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*540
if this polytope has a name.
Group : SmallGroup(540,54)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 9, 135, 45
Order of s₀s₁s₂ : 15
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,30,2} of size 1080
Vertex Figure Of :
   {2,6,30} of size 1080
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,30}*1080c
   3-fold covers : {6,90}*1620a, {18,30}*1620a, {6,30}*1620a, {6,30}*1620b, {6,90}*1620b, {18,30}*1620b, {6,90}*1620c, {18,30}*1620c
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)( 2,21)( 3,20)( 4,16)( 5,18)( 6,17)( 7,28)( 8,30)( 9,29)(10,25)
(11,27)(12,26)(13,22)(14,24)(15,23)(31,34)(32,36)(33,35)(37,43)(38,45)(39,44)
(41,42);;
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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)( 2,21)( 3,20)( 4,16)( 5,18)( 6,17)( 7,28)( 8,30)( 9,29)
(10,25)(11,27)(12,26)(13,22)(14,24)(15,23)(31,34)(32,36)(33,35)(37,43)(38,45)
(39,44)(41,42);
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope