Polytope of Type {4,10,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,10,6}*480
Also Known As : {{4,10|2},{10,6|2}}. if this polytope has another name.
Group : SmallGroup(480,1097)
Rank : 4
Schlafli Type : {4,10,6}
Number of vertices, edges, etc : 4, 20, 30, 6
Order of s₀s₁s₂s₃ : 60
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,10,6,2} of size 960
   {4,10,6,3} of size 1440
   {4,10,6,4} of size 1920
   {4,10,6,3} of size 1920
   {4,10,6,4} of size 1920
Vertex Figure Of :
   {2,4,10,6} of size 960
   {4,4,10,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,10,6}*240
   3-fold quotients : {4,10,2}*160
   5-fold quotients : {4,2,6}*96
   6-fold quotients : {2,10,2}*80
   10-fold quotients : {4,2,3}*48, {2,2,6}*48
   12-fold quotients : {2,5,2}*40
   15-fold quotients : {4,2,2}*32
   20-fold quotients : {2,2,3}*24
   30-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,20,6}*960, {4,10,12}*960, {8,10,6}*960
   3-fold covers : {4,10,18}*1440, {12,10,6}*1440, {4,30,6}*1440a, {4,30,6}*1440b
   4-fold covers : {4,20,12}*1920, {8,20,6}*1920a, {4,40,6}*1920a, {8,20,6}*1920b, {4,40,6}*1920b, {4,20,6}*1920a, {8,10,12}*1920, {4,10,24}*1920, {16,10,6}*1920, {4,20,6}*1920c
Permutation Representation (GAP) :

s0 := (31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)
(41,56)(42,57)(43,58)(44,59)(45,60);;
s1 := ( 1,31)( 2,35)( 3,34)( 4,33)( 5,32)( 6,36)( 7,40)( 8,39)( 9,38)(10,37)
(11,41)(12,45)(13,44)(14,43)(15,42)(16,46)(17,50)(18,49)(19,48)(20,47)(21,51)
(22,55)(23,54)(24,53)(25,52)(26,56)(27,60)(28,59)(29,58)(30,57);;
s2 := ( 1, 2)( 3, 5)( 6,12)( 7,11)( 8,15)( 9,14)(10,13)(16,17)(18,20)(21,27)
(22,26)(23,30)(24,29)(25,28)(31,32)(33,35)(36,42)(37,41)(38,45)(39,44)(40,43)
(46,47)(48,50)(51,57)(52,56)(53,60)(54,59)(55,58);;
s3 := ( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(16,21)(17,22)(18,23)(19,24)(20,25)
(31,36)(32,37)(33,38)(34,39)(35,40)(46,51)(47,52)(48,53)(49,54)(50,55);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(60)!(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)
(40,55)(41,56)(42,57)(43,58)(44,59)(45,60);
s1 := Sym(60)!( 1,31)( 2,35)( 3,34)( 4,33)( 5,32)( 6,36)( 7,40)( 8,39)( 9,38)
(10,37)(11,41)(12,45)(13,44)(14,43)(15,42)(16,46)(17,50)(18,49)(19,48)(20,47)
(21,51)(22,55)(23,54)(24,53)(25,52)(26,56)(27,60)(28,59)(29,58)(30,57);
s2 := Sym(60)!( 1, 2)( 3, 5)( 6,12)( 7,11)( 8,15)( 9,14)(10,13)(16,17)(18,20)
(21,27)(22,26)(23,30)(24,29)(25,28)(31,32)(33,35)(36,42)(37,41)(38,45)(39,44)
(40,43)(46,47)(48,50)(51,57)(52,56)(53,60)(54,59)(55,58);
s3 := Sym(60)!( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(16,21)(17,22)(18,23)(19,24)
(20,25)(31,36)(32,37)(33,38)(34,39)(35,40)(46,51)(47,52)(48,53)(49,54)(50,55);
poly := sub<Sym(60)|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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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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