Polytope of Type {4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*576
Also Known As : {4,4}_(6,6), {4,4}₁₂. if this polytope has another name.
Group : SmallGroup(576,5296)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 72, 144, 72
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   {4,4,2} of size 1152
Vertex Figure Of :
   {2,4,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*288
   4-fold quotients : {4,4}*144
   8-fold quotients : {4,4}*72
   9-fold quotients : {4,4}*64
   18-fold quotients : {4,4}*32
   36-fold quotients : {2,4}*16, {4,2}*16
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,8}*1152a, {8,4}*1152a, {4,8}*1152b, {8,4}*1152b, {4,4}*1152
   3-fold covers : {4,12}*1728b, {12,4}*1728a, {4,12}*1728d, {12,4}*1728d
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,34)
(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)
(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);;
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)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)(43,57)(44,60)(45,63)
(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)(54,72);;
s2 := ( 1,47)( 2,46)( 3,48)( 4,50)( 5,49)( 6,51)( 7,53)( 8,52)( 9,54)(10,38)
(11,37)(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,56)(20,55)(21,57)
(22,59)(23,58)(24,60)(25,62)(26,61)(27,63)(28,65)(29,64)(30,66)(31,68)(32,67)
(33,69)(34,71)(35,70)(36,72);;
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, 
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*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(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)
(22,34)(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)
(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);
s1 := Sym(72)!( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(20,22)(21,25)(24,26)
(29,31)(30,34)(33,35)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)(43,57)(44,60)
(45,63)(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)(54,72);
s2 := Sym(72)!( 1,47)( 2,46)( 3,48)( 4,50)( 5,49)( 6,51)( 7,53)( 8,52)( 9,54)
(10,38)(11,37)(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,56)(20,55)
(21,57)(22,59)(23,58)(24,60)(25,62)(26,61)(27,63)(28,65)(29,64)(30,66)(31,68)
(32,67)(33,69)(34,71)(35,70)(36,72);
poly := sub<Sym(72)|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, 
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope