Polytope of Type {2,4,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,10}*160
if this polytope has a name.
Group : SmallGroup(160,217)
Rank : 4
Schlafli Type : {2,4,10}
Number of vertices, edges, etc : 2, 4, 20, 10
Order of s₀s₁s₂s₃ : 20
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,10,2} of size 320
   {2,4,10,4} of size 640
   {2,4,10,5} of size 800
   {2,4,10,3} of size 960
   {2,4,10,5} of size 960
   {2,4,10,6} of size 960
   {2,4,10,8} of size 1280
   {2,4,10,10} of size 1600
   {2,4,10,10} of size 1600
   {2,4,10,10} of size 1600
   {2,4,10,12} of size 1920
   {2,4,10,4} of size 1920
   {2,4,10,6} of size 1920
   {2,4,10,3} of size 1920
   {2,4,10,5} of size 1920
   {2,4,10,6} of size 1920
   {2,4,10,6} of size 1920
   {2,4,10,10} of size 1920
   {2,4,10,10} of size 1920
Vertex Figure Of :
   {2,2,4,10} of size 320
   {3,2,4,10} of size 480
   {4,2,4,10} of size 640
   {5,2,4,10} of size 800
   {6,2,4,10} of size 960
   {7,2,4,10} of size 1120
   {8,2,4,10} of size 1280
   {9,2,4,10} of size 1440
   {10,2,4,10} of size 1600
   {11,2,4,10} of size 1760
   {12,2,4,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,10}*80
   4-fold quotients : {2,2,5}*40
   5-fold quotients : {2,4,2}*32
   10-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,4,20}*320, {4,4,10}*320, {2,8,10}*320
   3-fold covers : {2,12,10}*480, {6,4,10}*480, {2,4,30}*480a
   4-fold covers : {4,4,20}*640, {2,4,40}*640a, {2,4,20}*640, {2,4,40}*640b, {2,8,20}*640a, {2,8,20}*640b, {4,8,10}*640a, {8,4,10}*640a, {4,8,10}*640b, {8,4,10}*640b, {4,4,10}*640, {2,16,10}*640
   5-fold covers : {2,4,50}*800, {2,20,10}*800a, {10,4,10}*800, {2,20,10}*800c
   6-fold covers : {4,12,10}*960a, {12,4,10}*960, {6,4,20}*960, {2,24,10}*960, {6,8,10}*960, {2,12,20}*960, {2,4,60}*960a, {4,4,30}*960, {2,8,30}*960
   7-fold covers : {2,28,10}*1120, {14,4,10}*1120, {2,4,70}*1120
   8-fold covers : {4,8,10}*1280a, {8,4,10}*1280a, {2,8,20}*1280a, {2,4,40}*1280a, {8,8,10}*1280a, {8,8,10}*1280b, {8,8,10}*1280c, {2,8,40}*1280a, {2,8,40}*1280b, {2,8,40}*1280c, {8,8,10}*1280d, {2,8,40}*1280d, {8,4,20}*1280a, {4,4,40}*1280a, {8,4,20}*1280b, {4,4,40}*1280b, {4,8,20}*1280a, {4,4,20}*1280a, {4,4,20}*1280b, {4,8,20}*1280b, {4,8,20}*1280c, {4,8,20}*1280d, {4,16,10}*1280a, {16,4,10}*1280a, {2,16,20}*1280a, {2,4,80}*1280a, {4,16,10}*1280b, {16,4,10}*1280b, {2,16,20}*1280b, {2,4,80}*1280b, {4,4,10}*1280, {4,8,10}*1280b, {8,4,10}*1280b, {2,4,20}*1280a, {2,4,40}*1280b, {2,8,20}*1280b, {2,32,10}*1280
   9-fold covers : {2,36,10}*1440, {18,4,10}*1440, {2,4,90}*1440a, {6,12,10}*1440a, {6,12,10}*1440b, {2,12,30}*1440a, {6,12,10}*1440c, {2,12,30}*1440b, {6,4,30}*1440, {2,12,30}*1440c, {6,4,10}*1440c, {2,4,30}*1440
   10-fold covers : {2,4,100}*1600, {4,4,50}*1600, {2,8,50}*1600, {4,20,10}*1600a, {10,4,20}*1600, {20,4,10}*1600, {2,40,10}*1600a, {10,8,10}*1600, {2,20,20}*1600a, {2,20,20}*1600b, {2,40,10}*1600c, {4,20,10}*1600c
   11-fold covers : {2,44,10}*1760, {22,4,10}*1760, {2,4,110}*1760
   12-fold covers : {4,4,60}*1920, {4,12,20}*1920a, {12,4,20}*1920, {4,8,30}*1920a, {8,4,30}*1920a, {2,8,60}*1920a, {2,4,120}*1920a, {8,12,10}*1920a, {12,8,10}*1920a, {6,8,20}*1920a, {4,24,10}*1920a, {24,4,10}*1920a, {6,4,40}*1920a, {2,12,40}*1920a, {2,24,20}*1920a, {4,8,30}*1920b, {8,4,30}*1920b, {2,8,60}*1920b, {2,4,120}*1920b, {8,12,10}*1920b, {12,8,10}*1920b, {6,8,20}*1920b, {4,24,10}*1920b, {24,4,10}*1920b, {6,4,40}*1920b, {2,12,40}*1920b, {2,24,20}*1920b, {4,4,30}*1920a, {2,4,60}*1920a, {4,12,10}*1920a, {12,4,10}*1920a, {6,4,20}*1920a, {2,12,20}*1920a, {2,16,30}*1920, {6,16,10}*1920, {2,48,10}*1920, {4,12,10}*1920b, {2,12,20}*1920b, {6,4,10}*1920, {6,12,10}*1920a, {2,12,30}*1920b, {2,4,30}*1920b
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 7)( 8,13)( 9,14)(15,19)(16,20);;
s2 := ( 3, 4)( 5, 9)( 6, 8)( 7,12)(10,16)(11,15)(13,18)(14,17)(19,22)(20,21);;
s3 := ( 3, 5)( 4, 8)( 6,10)( 7,13)( 9,15)(12,17)(14,19)(18,21);;
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*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
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(22)!(1,2);
s1 := Sym(22)!( 4, 7)( 8,13)( 9,14)(15,19)(16,20);
s2 := Sym(22)!( 3, 4)( 5, 9)( 6, 8)( 7,12)(10,16)(11,15)(13,18)(14,17)(19,22)
(20,21);
s3 := Sym(22)!( 3, 5)( 4, 8)( 6,10)( 7,13)( 9,15)(12,17)(14,19)(18,21);
poly := sub<Sym(22)|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*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope