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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,10,4}*640
if this polytope has a name.
Group : SmallGroup(640,19900)
Rank : 5
Schlafli Type : {2,4,10,4}
Number of vertices, edges, etc : 2, 4, 20, 20, 4
Order of s₀s₁s₂s₃s₄ : 20
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,10,4,2} of size 1280
Vertex Figure Of :
   {2,2,4,10,4} of size 1280
   {3,2,4,10,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,10,4}*320, {2,4,10,2}*320
   4-fold quotients : {2,2,10,2}*160
   5-fold quotients : {2,4,2,4}*128
   8-fold quotients : {2,2,5,2}*80
   10-fold quotients : {2,2,2,4}*64, {2,4,2,2}*64
   20-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,4,20,4}*1280, {4,4,10,4}*1280, {2,4,10,8}*1280, {2,8,10,4}*1280
   3-fold covers : {2,4,30,4}*1920a, {6,4,10,4}*1920, {2,4,10,12}*1920, {2,12,10,4}*1920
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)
(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82);;
s2 := ( 3,43)( 4,47)( 5,46)( 6,45)( 7,44)( 8,48)( 9,52)(10,51)(11,50)(12,49)
(13,53)(14,57)(15,56)(16,55)(17,54)(18,58)(19,62)(20,61)(21,60)(22,59)(23,73)
(24,77)(25,76)(26,75)(27,74)(28,78)(29,82)(30,81)(31,80)(32,79)(33,63)(34,67)
(35,66)(36,65)(37,64)(38,68)(39,72)(40,71)(41,70)(42,69);;
s3 := ( 3, 4)( 5, 7)( 8, 9)(10,12)(13,14)(15,17)(18,19)(20,22)(23,39)(24,38)
(25,42)(26,41)(27,40)(28,34)(29,33)(30,37)(31,36)(32,35)(43,44)(45,47)(48,49)
(50,52)(53,54)(55,57)(58,59)(60,62)(63,79)(64,78)(65,82)(66,81)(67,80)(68,74)
(69,73)(70,77)(71,76)(72,75);;
s4 := ( 3,23)( 4,24)( 5,25)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(11,31)(12,32)
(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(43,73)
(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,63)(54,64)
(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
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(82)!(1,2);
s1 := Sym(82)!(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)
(52,62)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82);
s2 := Sym(82)!( 3,43)( 4,47)( 5,46)( 6,45)( 7,44)( 8,48)( 9,52)(10,51)(11,50)
(12,49)(13,53)(14,57)(15,56)(16,55)(17,54)(18,58)(19,62)(20,61)(21,60)(22,59)
(23,73)(24,77)(25,76)(26,75)(27,74)(28,78)(29,82)(30,81)(31,80)(32,79)(33,63)
(34,67)(35,66)(36,65)(37,64)(38,68)(39,72)(40,71)(41,70)(42,69);
s3 := Sym(82)!( 3, 4)( 5, 7)( 8, 9)(10,12)(13,14)(15,17)(18,19)(20,22)(23,39)
(24,38)(25,42)(26,41)(27,40)(28,34)(29,33)(30,37)(31,36)(32,35)(43,44)(45,47)
(48,49)(50,52)(53,54)(55,57)(58,59)(60,62)(63,79)(64,78)(65,82)(66,81)(67,80)
(68,74)(69,73)(70,77)(71,76)(72,75);
s4 := Sym(82)!( 3,23)( 4,24)( 5,25)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(11,31)
(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)
(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,63)
(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72);
poly := sub<Sym(82)|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*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, 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 >;

to this polytope