Polytope of Type {4,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,5}*720
Also Known As : {4,5|5}. if this polytope has another name.
Group : SmallGroup(720,764)
Rank : 3
Schlafli Type : {4,5}
Number of vertices, edges, etc : 72, 180, 90
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 5
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   {4,5,2} of size 1440
Vertex Figure Of :
   {2,4,5} of size 1440
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,5}*1440, {4,10}*1440e, {4,10}*1440f
Permutation Representation (GAP) :

s0 := ( 1, 3)( 2, 8)( 4,10)( 5, 6)( 7, 9);;
s1 := ( 1, 3)( 2, 4)( 5, 8)( 6, 9)( 7,10);;
s2 := ( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10);;
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*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(10)!( 1, 3)( 2, 8)( 4,10)( 5, 6)( 7, 9);
s1 := Sym(10)!( 1, 3)( 2, 4)( 5, 8)( 6, 9)( 7,10);
s2 := Sym(10)!( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10);
poly := sub<Sym(10)|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*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
to this polytope