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# Polytope of Type {10,15}

Atlas Canonical Name : {10,15}*360
if this polytope has a name.
Group : SmallGroup(360,121)
Rank : 3
Schlafli Type : {10,15}
Number of vertices, edges, etc : 12, 90, 18
Order of s0s1s2 : 3
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{10,15,2} of size 720
{10,15,4} of size 1440
Vertex Figure Of :
{2,10,15} of size 720
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {10,5}*120b
6-fold quotients : {5,5}*60
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,15}*720, {10,30}*720a, {10,30}*720b
3-fold covers : {10,45}*1080
4-fold covers : {10,60}*1440a, {10,60}*1440b, {20,15}*1440a, {20,15}*1440b, {10,30}*1440
5-fold covers : {10,15}*1800a
Permutation Representation (GAP) :
s0 := (5,6)(7,8);;
s1 := (2,3)(4,5)(6,7);;
s2 := (1,2)(5,7)(6,8);;
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*s0*s1*s2*s0*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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(8)!(5,6)(7,8);
s1 := Sym(8)!(2,3)(4,5)(6,7);
s2 := Sym(8)!(1,2)(5,7)(6,8);
poly := sub<Sym(8)|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*s0*s1*s2*s0*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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