Polytope of Type {10,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,4}*320
Also Known As : {{10,4|2},{4,4|2}}. if this polytope has another name.
Group : SmallGroup(320,1260)
Rank : 4
Schlafli Type : {10,4,4}
Number of vertices, edges, etc : 10, 20, 8, 4
Order of s₀s₁s₂s₃ : 20
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,4,4,2} of size 640
   {10,4,4,4} of size 1280
   {10,4,4,6} of size 1920
   {10,4,4,3} of size 1920
Vertex Figure Of :
   {2,10,4,4} of size 640
   {4,10,4,4} of size 1280
   {5,10,4,4} of size 1600
   {6,10,4,4} of size 1920
   {3,10,4,4} of size 1920
   {5,10,4,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,2,4}*160, {10,4,2}*160
   4-fold quotients : {5,2,4}*80, {10,2,2}*80
   5-fold quotients : {2,4,4}*64
   8-fold quotients : {5,2,2}*40
   10-fold quotients : {2,2,4}*32, {2,4,2}*32
   20-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,4,4}*640, {10,4,8}*640a, {10,8,4}*640a, {10,4,8}*640b, {10,8,4}*640b, {10,4,4}*640
   3-fold covers : {10,4,12}*960, {10,12,4}*960a, {30,4,4}*960
   4-fold covers : {10,4,8}*1280a, {10,8,4}*1280a, {10,8,8}*1280a, {10,8,8}*1280b, {10,8,8}*1280c, {10,8,8}*1280d, {20,4,8}*1280a, {40,4,4}*1280a, {20,4,8}*1280b, {40,4,4}*1280b, {20,8,4}*1280a, {20,4,4}*1280a, {20,4,4}*1280b, {20,8,4}*1280b, {20,8,4}*1280c, {20,8,4}*1280d, {10,4,16}*1280a, {10,16,4}*1280a, {10,4,16}*1280b, {10,16,4}*1280b, {10,4,4}*1280, {10,4,8}*1280b, {10,8,4}*1280b
   5-fold covers : {50,4,4}*1600, {10,4,20}*1600, {10,20,4}*1600a, {10,20,4}*1600c
   6-fold covers : {60,4,4}*1920, {20,12,4}*1920a, {20,4,12}*1920, {30,4,8}*1920a, {30,8,4}*1920a, {10,8,12}*1920a, {10,12,8}*1920a, {10,4,24}*1920a, {10,24,4}*1920a, {30,4,8}*1920b, {30,8,4}*1920b, {10,8,12}*1920b, {10,12,8}*1920b, {10,4,24}*1920b, {10,24,4}*1920b, {30,4,4}*1920a, {10,4,12}*1920a, {10,12,4}*1920a
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)(11,12)(13,15)(16,17)(18,20)(21,27)(22,26)
(23,30)(24,29)(25,28)(31,37)(32,36)(33,40)(34,39)(35,38)(41,42)(43,45)(46,47)
(48,50)(51,52)(53,55)(56,57)(58,60)(61,67)(62,66)(63,70)(64,69)(65,68)(71,77)
(72,76)(73,80)(74,79)(75,78);;
s2 := ( 1,21)( 2,22)( 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)(41,61)
(42,62)(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);;
s3 := ( 1,41)( 2,42)( 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)(21,71)
(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,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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(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)(11,12)(13,15)(16,17)(18,20)(21,27)
(22,26)(23,30)(24,29)(25,28)(31,37)(32,36)(33,40)(34,39)(35,38)(41,42)(43,45)
(46,47)(48,50)(51,52)(53,55)(56,57)(58,60)(61,67)(62,66)(63,70)(64,69)(65,68)
(71,77)(72,76)(73,80)(74,79)(75,78);
s2 := Sym(80)!( 1,21)( 2,22)( 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)
(41,61)(42,62)(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);
s3 := Sym(80)!( 1,41)( 2,42)( 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)
(21,71)(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,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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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