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Polytope of Type {2,3,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,5}*120
if this polytope has a name.
Group : SmallGroup(120,35)
Rank : 4
Schlafli Type : {2,3,5}
Number of vertices, edges, etc : 2, 6, 15, 10
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Locally Projective
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,3,5,2} of size 240
   {2,3,5,3} of size 1320
   {2,3,5,4} of size 1920
Vertex Figure Of :
   {2,2,3,5} of size 240
   {3,2,3,5} of size 360
   {4,2,3,5} of size 480
   {5,2,3,5} of size 600
   {6,2,3,5} of size 720
   {7,2,3,5} of size 840
   {8,2,3,5} of size 960
   {9,2,3,5} of size 1080
   {10,2,3,5} of size 1200
   {11,2,3,5} of size 1320
   {12,2,3,5} of size 1440
   {13,2,3,5} of size 1560
   {14,2,3,5} of size 1680
   {15,2,3,5} of size 1800
   {16,2,3,5} of size 1920
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,3,5}*240, {2,3,10}*240a, {2,3,10}*240b, {2,6,5}*240b, {2,6,5}*240c
   4-fold covers : {4,6,5}*480b, {2,3,10}*480, {2,6,5}*480b, {2,6,10}*480c, {2,6,10}*480d, {2,6,10}*480e, {2,6,10}*480f
   6-fold covers : {2,3,10}*720, {6,6,5}*720b, {2,3,15}*720, {2,6,15}*720
   8-fold covers : {8,6,5}*960b, {4,6,5}*960b, {4,6,10}*960c, {4,6,10}*960d, {2,6,20}*960a, {2,6,20}*960b, {2,12,10}*960c, {2,12,10}*960d, {2,3,20}*960, {2,12,5}*960, {2,6,10}*960c
   10-fold covers : {2,6,5}*1200, {2,15,5}*1200, {10,6,5}*1200b, {2,15,10}*1200
   12-fold covers : {12,6,5}*1440b, {6,6,5}*1440b, {6,6,10}*1440c, {6,6,10}*1440d, {2,3,10}*1440b, {2,3,30}*1440, {2,6,10}*1440b, {2,6,10}*1440c, {2,6,15}*1440c, {2,6,15}*1440d, {2,6,30}*1440a, {2,6,30}*1440b
   14-fold covers : {2,6,35}*1680, {2,21,10}*1680, {14,6,5}*1680b
   16-fold covers : {16,6,5}*1920b, {4,12,10}*1920f, {4,12,10}*1920g, {8,6,5}*1920b, {2,6,40}*1920d, {8,6,10}*1920e, {2,6,40}*1920e, {8,6,10}*1920f, {2,24,10}*1920c, {2,24,10}*1920d, {4,6,10}*1920d, {2,6,20}*1920c, {2,12,10}*1920c, {4,12,5}*1920, {2,6,20}*1920e, {2,12,10}*1920e, {2,6,10}*1920b, {4,3,5}*1920, {2,6,5}*1920
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (4,5)(6,7);;
s2 := (3,4)(6,7);;
s3 := (4,6)(5,7);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(7)!(1,2);
s1 := Sym(7)!(4,5)(6,7);
s2 := Sym(7)!(3,4)(6,7);
s3 := Sym(7)!(4,6)(5,7);
poly := sub<Sym(7)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3 >; 
 

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