Polytope of Type {2,72,2,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,72,2,3}*1728
if this polytope has a name.
Group : SmallGroup(1728,15813)
Rank : 5
Schlafli Type : {2,72,2,3}
Number of vertices, edges, etc : 2, 72, 72, 3, 3
Order of s₀s₁s₂s₃s₄ : 72
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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) :
   2-fold quotients : {2,36,2,3}*864
   3-fold quotients : {2,24,2,3}*576
   4-fold quotients : {2,18,2,3}*432
   6-fold quotients : {2,12,2,3}*288
   8-fold quotients : {2,9,2,3}*216
   9-fold quotients : {2,8,2,3}*192
   12-fold quotients : {2,6,2,3}*144
   18-fold quotients : {2,4,2,3}*96
   24-fold quotients : {2,3,2,3}*72
   36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 5)( 6,10)( 7, 9)( 8,11)(13,14)(15,19)(16,18)(17,20)(21,30)(22,32)
(23,31)(24,37)(25,36)(26,38)(27,34)(28,33)(29,35)(39,57)(40,59)(41,58)(42,64)
(43,63)(44,65)(45,61)(46,60)(47,62)(48,66)(49,68)(50,67)(51,73)(52,72)(53,74)
(54,70)(55,69)(56,71);;
s2 := ( 3,42)( 4,44)( 5,43)( 6,39)( 7,41)( 8,40)( 9,46)(10,45)(11,47)(12,51)
(13,53)(14,52)(15,48)(16,50)(17,49)(18,55)(19,54)(20,56)(21,69)(22,71)(23,70)
(24,66)(25,68)(26,67)(27,73)(28,72)(29,74)(30,60)(31,62)(32,61)(33,57)(34,59)
(35,58)(36,64)(37,63)(38,65);;
s3 := (76,77);;
s4 := (75,76);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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(77)!(1,2);
s1 := Sym(77)!( 4, 5)( 6,10)( 7, 9)( 8,11)(13,14)(15,19)(16,18)(17,20)(21,30)
(22,32)(23,31)(24,37)(25,36)(26,38)(27,34)(28,33)(29,35)(39,57)(40,59)(41,58)
(42,64)(43,63)(44,65)(45,61)(46,60)(47,62)(48,66)(49,68)(50,67)(51,73)(52,72)
(53,74)(54,70)(55,69)(56,71);
s2 := Sym(77)!( 3,42)( 4,44)( 5,43)( 6,39)( 7,41)( 8,40)( 9,46)(10,45)(11,47)
(12,51)(13,53)(14,52)(15,48)(16,50)(17,49)(18,55)(19,54)(20,56)(21,69)(22,71)
(23,70)(24,66)(25,68)(26,67)(27,73)(28,72)(29,74)(30,60)(31,62)(32,61)(33,57)
(34,59)(35,58)(36,64)(37,63)(38,65);
s3 := Sym(77)!(76,77);
s4 := Sym(77)!(75,76);
poly := sub<Sym(77)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope