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# Polytope of Type {5,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,5}*120
Also Known As : {5,5|3}. if this polytope has another name.
Group : SmallGroup(120,35)
Rank : 3
Schlafli Type : {5,5}
Number of vertices, edges, etc : 12, 30, 12
Order of s0s1s2 : 6
Order of s0s1s2s1 : 3
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{5,5,2} of size 240
Vertex Figure Of :
{2,5,5} of size 240
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {5,5}*60
Covers (Minimal Covers in Boldface) :
2-fold covers : {5,10}*240, {10,5}*240
4-fold covers : {5,20}*480, {20,5}*480, {10,10}*480
5-fold covers : {5,5}*600
6-fold covers : {10,15}*720, {15,10}*720
8-fold covers : {10,20}*960a, {20,10}*960a, {10,20}*960b, {20,10}*960b, {10,10}*960
10-fold covers : {5,10}*1200a, {5,10}*1200b, {10,5}*1200a, {10,5}*1200b
12-fold covers : {15,20}*1440a, {20,15}*1440a, {15,15}*1440, {10,30}*1440, {30,10}*1440
14-fold covers : {10,35}*1680, {35,10}*1680
16-fold covers : {20,20}*1920a, {10,40}*1920a, {40,10}*1920a, {10,20}*1920, {20,10}*1920, {20,20}*1920b, {20,20}*1920c, {20,20}*1920d, {10,40}*1920b, {40,10}*1920b, {5,5}*1920a
Permutation Representation (GAP) :
s0 := ( 2, 9)( 4,12)( 5, 7)( 6, 8);;
s1 := ( 3, 5)( 4,11)( 6,12)( 7, 9);;
s2 := ( 1,11)( 3,10)( 4, 7)( 5,12);;
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*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(12)!( 2, 9)( 4,12)( 5, 7)( 6, 8);
s1 := Sym(12)!( 3, 5)( 4,11)( 6,12)( 7, 9);
s2 := Sym(12)!( 1,11)( 3,10)( 4, 7)( 5,12);
poly := sub<Sym(12)|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*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
to this polytope