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# Polytope of Type {8,10}

Atlas Canonical Name : {8,10}*720a
if this polytope has a name.
Group : SmallGroup(720,764)
Rank : 3
Schlafli Type : {8,10}
Number of vertices, edges, etc : 36, 180, 45
Order of s0s1s2 : 10
Order of s0s1s2s1 : 3
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{8,10,2} of size 1440
Vertex Figure Of :
{2,8,10} of size 1440
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,10}*1440c, {8,10}*1440e, {8,10}*1440f
Permutation Representation (GAP) :
s0 := (2,8)(3,5)(4,7)(6,9);;
s1 := ( 1, 2)( 3, 5)( 4, 9)( 6,10)( 7, 8);;
s2 := ( 1,10)( 3, 4)( 5, 7)( 6, 9);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;

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

References : None.
to this polytope