Polytope of Type {8,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,10}*160
Also Known As : {8,10|2}. if this polytope has another name.
Group : SmallGroup(160,131)
Rank : 3
Schlafli Type : {8,10}
Number of vertices, edges, etc : 8, 40, 10
Order of s₀s₁s₂ : 40
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,10,2} of size 320
   {8,10,4} of size 640
   {8,10,5} of size 800
   {8,10,3} of size 960
   {8,10,5} of size 960
   {8,10,6} of size 960
   {8,10,8} of size 1280
   {8,10,10} of size 1600
   {8,10,10} of size 1600
   {8,10,10} of size 1600
   {8,10,12} of size 1920
   {8,10,4} of size 1920
   {8,10,6} of size 1920
   {8,10,3} of size 1920
   {8,10,5} of size 1920
   {8,10,6} of size 1920
   {8,10,6} of size 1920
   {8,10,10} of size 1920
   {8,10,10} of size 1920
Vertex Figure Of :
   {2,8,10} of size 320
   {4,8,10} of size 640
   {4,8,10} of size 640
   {6,8,10} of size 960
   {3,8,10} of size 960
   {4,8,10} of size 1280
   {8,8,10} of size 1280
   {8,8,10} of size 1280
   {8,8,10} of size 1280
   {8,8,10} of size 1280
   {4,8,10} of size 1280
   {10,8,10} of size 1600
   {12,8,10} of size 1920
   {12,8,10} of size 1920
   {3,8,10} of size 1920
   {6,8,10} of size 1920
   {6,8,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,10}*80
   4-fold quotients : {2,10}*40
   5-fold quotients : {8,2}*32
   8-fold quotients : {2,5}*20
   10-fold quotients : {4,2}*16
   20-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,20}*320a, {16,10}*320
   3-fold covers : {24,10}*480, {8,30}*480
   4-fold covers : {8,40}*640b, {8,20}*640a, {8,40}*640d, {16,20}*640a, {16,20}*640b, {32,10}*640
   5-fold covers : {8,50}*800, {40,10}*800a, {40,10}*800c
   6-fold covers : {48,10}*960, {24,20}*960a, {8,60}*960a, {16,30}*960
   7-fold covers : {56,10}*1120, {8,70}*1120
   8-fold covers : {8,40}*1280a, {8,20}*1280a, {8,40}*1280c, {16,20}*1280a, {16,20}*1280b, {8,80}*1280a, {8,80}*1280b, {16,40}*1280c, {8,80}*1280d, {16,40}*1280d, {16,40}*1280e, {8,80}*1280f, {16,40}*1280f, {32,20}*1280a, {32,20}*1280b, {64,10}*1280
   9-fold covers : {72,10}*1440, {8,90}*1440, {24,30}*1440a, {24,30}*1440b, {24,30}*1440c, {8,30}*1440
   10-fold covers : {8,100}*1600a, {16,50}*1600, {80,10}*1600a, {40,20}*1600b, {40,20}*1600c, {80,10}*1600c
   11-fold covers : {88,10}*1760, {8,110}*1760
   12-fold covers : {8,60}*1920a, {24,20}*1920a, {8,120}*1920a, {8,120}*1920c, {24,40}*1920a, {24,40}*1920b, {16,60}*1920a, {48,20}*1920a, {16,60}*1920b, {48,20}*1920b, {32,30}*1920, {96,10}*1920, {24,20}*1920c, {24,30}*1920a, {8,30}*1920g
Permutation Representation (GAP) :

s0 := (11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)
(26,31)(27,32)(28,33)(29,34)(30,35);;
s1 := ( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)
(11,36)(12,40)(13,39)(14,38)(15,37)(16,31)(17,35)(18,34)(19,33)(20,32);;
s2 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)
(26,27)(28,30)(31,32)(33,35)(36,37)(38,40);;
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*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(40)!(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)
(25,40)(26,31)(27,32)(28,33)(29,34)(30,35);
s1 := Sym(40)!( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,26)( 7,30)( 8,29)( 9,28)
(10,27)(11,36)(12,40)(13,39)(14,38)(15,37)(16,31)(17,35)(18,34)(19,33)(20,32);
s2 := Sym(40)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)
(23,25)(26,27)(28,30)(31,32)(33,35)(36,37)(38,40);
poly := sub<Sym(40)|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*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope