Polytope of Type {40}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {40}*80
Also Known As : 40-gon, {40}. if this polytope has another name.
Group : SmallGroup(80,7)
Rank : 2
Schlafli Type : {40}
Number of vertices, edges, etc : 40, 40
Order of s0s1 : 40
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {40,2} of size 160
   {40,4} of size 320
   {40,4} of size 320
   {40,6} of size 480
   {40,4} of size 640
   {40,8} of size 640
   {40,8} of size 640
   {40,8} of size 640
   {40,8} of size 640
   {40,4} of size 640
   {40,10} of size 800
   {40,10} of size 800
   {40,10} of size 800
   {40,12} of size 960
   {40,12} of size 960
   {40,6} of size 960
   {40,6} of size 960
   {40,10} of size 960
   {40,10} of size 960
   {40,6} of size 960
   {40,6} of size 960
   {40,6} of size 960
   {40,14} of size 1120
   {40,8} of size 1280
   {40,8} of size 1280
   {40,4} of size 1280
   {40,8} of size 1280
   {40,8} of size 1280
   {40,16} of size 1280
   {40,16} of size 1280
   {40,16} of size 1280
   {40,16} of size 1280
   {40,16} of size 1280
   {40,16} of size 1280
   {40,4} of size 1280
   {40,8} of size 1280
   {40,4} of size 1280
   {40,4} of size 1280
   {40,8} of size 1280
   {40,8} of size 1280
   {40,8} of size 1280
   {40,4} of size 1280
   {40,4} of size 1280
   {40,4} of size 1280
   {40,4} of size 1280
   {40,18} of size 1440
   {40,6} of size 1440
   {40,20} of size 1600
   {40,20} of size 1600
   {40,20} of size 1600
   {40,20} of size 1600
   {40,20} of size 1600
   {40,20} of size 1600
   {40,4} of size 1600
   {40,4} of size 1600
   {40,22} of size 1760
   {40,12} of size 1920
   {40,24} of size 1920
   {40,24} of size 1920
   {40,24} of size 1920
   {40,24} of size 1920
   {40,12} of size 1920
   {40,12} of size 1920
   {40,12} of size 1920
   {40,6} of size 1920
   {40,12} of size 1920
   {40,12} of size 1920
   {40,6} of size 1920
   {40,6} of size 1920
   {40,6} of size 1920
   {40,12} of size 1920
   {40,12} of size 1920
   {40,4} of size 1920
   {40,6} of size 1920
   {40,6} of size 1920
   {40,10} of size 1920
   {40,4} of size 1920
   {40,4} of size 1920
   {40,4} of size 1920
   {40,6} of size 1920
   {40,6} of size 1920
   {40,10} of size 1920
Vertex Figure Of :
   {2,40} of size 160
   {4,40} of size 320
   {4,40} of size 320
   {6,40} of size 480
   {4,40} of size 640
   {8,40} of size 640
   {8,40} of size 640
   {8,40} of size 640
   {8,40} of size 640
   {4,40} of size 640
   {10,40} of size 800
   {10,40} of size 800
   {10,40} of size 800
   {12,40} of size 960
   {12,40} of size 960
   {6,40} of size 960
   {6,40} of size 960
   {10,40} of size 960
   {10,40} of size 960
   {6,40} of size 960
   {6,40} of size 960
   {6,40} of size 960
   {14,40} of size 1120
   {8,40} of size 1280
   {8,40} of size 1280
   {4,40} of size 1280
   {8,40} of size 1280
   {8,40} of size 1280
   {16,40} of size 1280
   {16,40} of size 1280
   {16,40} of size 1280
   {16,40} of size 1280
   {16,40} of size 1280
   {16,40} of size 1280
   {4,40} of size 1280
   {8,40} of size 1280
   {4,40} of size 1280
   {4,40} of size 1280
   {8,40} of size 1280
   {8,40} of size 1280
   {8,40} of size 1280
   {4,40} of size 1280
   {4,40} of size 1280
   {4,40} of size 1280
   {4,40} of size 1280
   {18,40} of size 1440
   {6,40} of size 1440
   {20,40} of size 1600
   {20,40} of size 1600
   {20,40} of size 1600
   {20,40} of size 1600
   {20,40} of size 1600
   {20,40} of size 1600
   {4,40} of size 1600
   {4,40} of size 1600
   {22,40} of size 1760
   {12,40} of size 1920
   {24,40} of size 1920
   {24,40} of size 1920
   {24,40} of size 1920
   {24,40} of size 1920
   {12,40} of size 1920
   {12,40} of size 1920
   {12,40} of size 1920
   {6,40} of size 1920
   {12,40} of size 1920
   {12,40} of size 1920
   {6,40} of size 1920
   {6,40} of size 1920
   {6,40} of size 1920
   {12,40} of size 1920
   {12,40} of size 1920
   {4,40} of size 1920
   {6,40} of size 1920
   {6,40} of size 1920
   {10,40} of size 1920
   {4,40} of size 1920
   {4,40} of size 1920
   {4,40} of size 1920
   {6,40} of size 1920
   {6,40} of size 1920
   {10,40} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {20}*40
   4-fold quotients : {10}*20
   5-fold quotients : {8}*16
   8-fold quotients : {5}*10
   10-fold quotients : {4}*8
   20-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {80}*160
   3-fold covers : {120}*240
   4-fold covers : {160}*320
   5-fold covers : {200}*400
   6-fold covers : {240}*480
   7-fold covers : {280}*560
   8-fold covers : {320}*640
   9-fold covers : {360}*720
   10-fold covers : {400}*800
   11-fold covers : {440}*880
   12-fold covers : {480}*960
   13-fold covers : {520}*1040
   14-fold covers : {560}*1120
   15-fold covers : {600}*1200
   16-fold covers : {640}*1280
   17-fold covers : {680}*1360
   18-fold covers : {720}*1440
   19-fold covers : {760}*1520
   20-fold covers : {800}*1600
   21-fold covers : {840}*1680
   22-fold covers : {880}*1760
   23-fold covers : {920}*1840
   24-fold covers : {960}*1920
   25-fold covers : {1000}*2000
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 5)( 6, 9)( 7,11)( 8,10)(12,13)(14,19)(15,21)(16,20)(17,23)
(18,22)(25,30)(26,29)(27,32)(28,31)(33,34)(35,38)(36,37)(39,40);;
s1 := ( 1, 7)( 2, 4)( 3,15)( 5,17)( 6,10)( 8,12)( 9,25)(11,27)(13,18)(14,20)
(16,22)(19,33)(21,35)(23,28)(24,29)(26,31)(30,39)(32,36)(34,37)(38,40);;
poly := Group([s0,s1]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(40)!( 2, 3)( 4, 5)( 6, 9)( 7,11)( 8,10)(12,13)(14,19)(15,21)(16,20)
(17,23)(18,22)(25,30)(26,29)(27,32)(28,31)(33,34)(35,38)(36,37)(39,40);
s1 := Sym(40)!( 1, 7)( 2, 4)( 3,15)( 5,17)( 6,10)( 8,12)( 9,25)(11,27)(13,18)
(14,20)(16,22)(19,33)(21,35)(23,28)(24,29)(26,31)(30,39)(32,36)(34,37)(38,40);
poly := sub<Sym(40)|s0,s1>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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