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Polytope of Type {10,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4}*720
if this polytope has a name.
Group : SmallGroup(720,764)
Rank : 3
Schlafli Type : {10,4}
Number of vertices, edges, etc : 90, 180, 36
Order of s₀s₁s₂ : 10
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {10,4,2} of size 1440
Vertex Figure Of :
   {2,10,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {10,4}*1440b, {10,4}*1440c, {10,4}*1440d
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope