Polytope of Type {3,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,10}*1080
Also Known As : {{3,6}₆,{6,10|2}}. if this polytope has another name.
Group : SmallGroup(1080,287)
Rank : 4
Schlafli Type : {3,6,10}
Number of vertices, edges, etc : 9, 27, 90, 10
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
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 : {3,6,10}*360
   5-fold quotients : {3,6,2}*216
   9-fold quotients : {3,2,10}*120
   15-fold quotients : {3,6,2}*72
   18-fold quotients : {3,2,5}*60
   45-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)
(26,41)(27,42)(28,43)(29,44)(30,45);;
s1 := ( 1,17)( 2,18)( 3,16)( 4,20)( 5,21)( 6,19)( 7,23)( 8,24)( 9,22)(10,26)
(11,27)(12,25)(13,29)(14,30)(15,28);;
s2 := ( 2, 3)( 4,13)( 5,15)( 6,14)( 7,10)( 8,12)( 9,11)(17,18)(19,28)(20,30)
(21,29)(22,25)(23,27)(24,26)(32,33)(34,43)(35,45)(36,44)(37,40)(38,42)
(39,41);;
s3 := ( 1, 4)( 2, 5)( 3, 6)( 7,13)( 8,14)( 9,15)(16,19)(17,20)(18,21)(22,28)
(23,29)(24,30)(31,34)(32,35)(33,36)(37,43)(38,44)(39,45);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(45)!(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)
(25,40)(26,41)(27,42)(28,43)(29,44)(30,45);
s1 := Sym(45)!( 1,17)( 2,18)( 3,16)( 4,20)( 5,21)( 6,19)( 7,23)( 8,24)( 9,22)
(10,26)(11,27)(12,25)(13,29)(14,30)(15,28);
s2 := Sym(45)!( 2, 3)( 4,13)( 5,15)( 6,14)( 7,10)( 8,12)( 9,11)(17,18)(19,28)
(20,30)(21,29)(22,25)(23,27)(24,26)(32,33)(34,43)(35,45)(36,44)(37,40)(38,42)
(39,41);
s3 := Sym(45)!( 1, 4)( 2, 5)( 3, 6)( 7,13)( 8,14)( 9,15)(16,19)(17,20)(18,21)
(22,28)(23,29)(24,30)(31,34)(32,35)(33,36)(37,43)(38,44)(39,45);
poly := sub<Sym(45)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope