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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,10,4,4}*640
if this polytope has a name.
Group : SmallGroup(640,19898)
Rank : 5
Schlafli Type : {2,10,4,4}
Number of vertices, edges, etc : 2, 10, 20, 8, 4
Order of s₀s₁s₂s₃s₄ : 20
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,10,4,4,2} of size 1280
Vertex Figure Of :
   {2,2,10,4,4} of size 1280
   {3,2,10,4,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,10,2,4}*320, {2,10,4,2}*320
   4-fold quotients : {2,5,2,4}*160, {2,10,2,2}*160
   5-fold quotients : {2,2,4,4}*128
   8-fold quotients : {2,5,2,2}*80
   10-fold quotients : {2,2,2,4}*64, {2,2,4,2}*64
   20-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,20,4,4}*1280, {4,10,4,4}*1280, {2,10,4,8}*1280a, {2,10,8,4}*1280a, {2,10,4,8}*1280b, {2,10,8,4}*1280b, {2,10,4,4}*1280
   3-fold covers : {2,30,4,4}*1920, {6,10,4,4}*1920, {2,10,4,12}*1920, {2,10,12,4}*1920a
Permutation Representation (GAP) :

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

to this polytope