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Polytope of Type {4,2,10,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,2,10,4}*640
if this polytope has a name.
Group : SmallGroup(640,20602)
Rank : 5
Schlafli Type : {4,2,10,4}
Number of vertices, edges, etc : 4, 4, 10, 20, 4
Order of s0s1s2s3s4 : 20
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,2,10,4,2} of size 1280
Vertex Figure Of :
   {2,4,2,10,4} of size 1280
   {3,4,2,10,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,10,4}*320, {4,2,10,2}*320
   4-fold quotients : {4,2,5,2}*160, {2,2,10,2}*160
   5-fold quotients : {4,2,2,4}*128
   8-fold quotients : {2,2,5,2}*80
   10-fold quotients : {2,2,2,4}*64, {4,2,2,2}*64
   20-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4,10,4}*1280, {4,2,20,4}*1280, {4,2,10,8}*1280, {8,2,10,4}*1280
   3-fold covers : {4,2,30,4}*1920a, {4,6,10,4}*1920a, {4,2,10,12}*1920, {12,2,10,4}*1920
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2)(3,4);;
s2 := ( 7, 8)(10,11)(12,13)(15,16)(17,18)(19,20)(21,22)(23,24);;
s3 := ( 5, 7)( 6,15)( 8,12)( 9,10)(11,21)(14,19)(16,17)(18,22)(20,23);;
s4 := ( 5, 6)( 7,10)( 8,11)( 9,14)(12,17)(13,18)(15,19)(16,20)(21,23)(22,24);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(24)!(2,3);
s1 := Sym(24)!(1,2)(3,4);
s2 := Sym(24)!( 7, 8)(10,11)(12,13)(15,16)(17,18)(19,20)(21,22)(23,24);
s3 := Sym(24)!( 5, 7)( 6,15)( 8,12)( 9,10)(11,21)(14,19)(16,17)(18,22)(20,23);
s4 := Sym(24)!( 5, 6)( 7,10)( 8,11)( 9,14)(12,17)(13,18)(15,19)(16,20)(21,23)
(22,24);
poly := sub<Sym(24)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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