Polytope of Type {10,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,16}*320
Also Known As : {10,16|2}. if this polytope has another name.
Group : SmallGroup(320,537)
Rank : 3
Schlafli Type : {10,16}
Number of vertices, edges, etc : 10, 80, 16
Order of s₀s₁s₂ : 80
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,16,2} of size 640
   {10,16,4} of size 1280
   {10,16,4} of size 1280
   {10,16,6} of size 1920
Vertex Figure Of :
   {2,10,16} of size 640
   {4,10,16} of size 1280
   {5,10,16} of size 1600
   {6,10,16} of size 1920
   {3,10,16} of size 1920
   {5,10,16} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,8}*160
   4-fold quotients : {10,4}*80
   5-fold quotients : {2,16}*64
   8-fold quotients : {10,2}*40
   10-fold quotients : {2,8}*32
   16-fold quotients : {5,2}*20
   20-fold quotients : {2,4}*16
   40-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,16}*640a, {10,32}*640
   3-fold covers : {10,48}*960, {30,16}*960
   4-fold covers : {20,16}*1280a, {40,16}*1280c, {40,16}*1280d, {20,32}*1280a, {20,32}*1280b, {10,64}*1280
   5-fold covers : {50,16}*1600, {10,80}*1600a, {10,80}*1600c
   6-fold covers : {60,16}*1920a, {20,48}*1920a, {30,32}*1920, {10,96}*1920
Permutation Representation (GAP) :

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

References : None.
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