Polytope of Type {3,10,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,10,2}*720
if this polytope has a name.
Group : SmallGroup(720,771)
Rank : 4
Schlafli Type : {3,10,2}
Number of vertices, edges, etc : 18, 90, 60, 2
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,10,2,2} of size 1440
Vertex Figure Of :
   {2,3,10,2} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,10,2}*240a
   6-fold quotients : {3,5,2}*120
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,10,2}*1440b, {6,10,2}*1440b, {6,10,2}*1440c
Permutation Representation (GAP) :

s0 := (2,3)(5,6)(7,8);;
s1 := (1,2)(4,5)(7,8);;
s2 := (5,7)(6,8);;
s3 := ( 9,10);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(10)!(2,3)(5,6)(7,8);
s1 := Sym(10)!(1,2)(4,5)(7,8);
s2 := Sym(10)!(5,7)(6,8);
s3 := Sym(10)!( 9,10);
poly := sub<Sym(10)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

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

to this polytope