Polytope of Type {4,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,4}*576a
Also Known As : ₂T⁴_(3,3)(2,0), {{4,4}₆,{4,4|2}}. if this polytope has another name.
Group : SmallGroup(576,8399)
Rank : 4
Schlafli Type : {4,4,4}
Number of vertices, edges, etc : 18, 36, 36, 4
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Locally Toroidal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,4,4,2} of size 1152
Vertex Figure Of :
   {2,4,4,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4,2}*288
   4-fold quotients : {4,4,2}*144
   18-fold quotients : {2,2,4}*32
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4,4}*1152b, {4,4,8}*1152
   3-fold covers : {4,12,4}*1728b, {12,4,4}*1728a, {4,4,12}*1728b, {4,12,4}*1728c, {12,4,4}*1728c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References :

Theorem 10C2, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambr\ idge University Press, 2002)

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