Polytope of Type {6,3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,6}*864b
Also Known As : ₆T⁴_(2,2)(1,1). if this polytope has another name.
Group : SmallGroup(864,4673)
Rank : 4
Schlafli Type : {6,3,6}
Number of vertices, edges, etc : 24, 36, 36, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Locally Toroidal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,3,6,2} of size 1728
Vertex Figure Of :
   {2,6,3,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,3,2}*288
   4-fold quotients : {6,3,6}*216
   9-fold quotients : {6,3,2}*96
   12-fold quotients : {2,3,6}*72, {6,3,2}*72
   18-fold quotients : {3,3,2}*48
   36-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,3,6}*1728, {6,6,6}*1728e
Permutation Representation (GAP) :

s0 := ( 2, 3)( 6, 7)(10,11)(14,15)(18,19)(22,23)(26,27)(30,31)(34,35);;
s1 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)(17,33)
(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);;
s2 := ( 1,20)( 2,18)( 3,19)( 4,17)( 5,16)( 6,14)( 7,15)( 8,13)( 9,24)(10,22)
(11,23)(12,21)(25,32)(26,30)(27,31)(28,29)(33,36);;
s3 := ( 5, 9)( 6,10)( 7,11)( 8,12)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)
(31,35)(32,36);;
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, s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References :

Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (\ Cambridge University Press, 2002)

to this polytope