Polytope of Type {4,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,10}*480a
Also Known As : {{4,6|2},{6,10|2}}. if this polytope has another name.
Group : SmallGroup(480,1097)
Rank : 4
Schlafli Type : {4,6,10}
Number of vertices, edges, etc : 4, 12, 30, 10
Order of s₀s₁s₂s₃ : 60
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,10,2} of size 960
   {4,6,10,4} of size 1920
Vertex Figure Of :
   {2,4,6,10} of size 960
   {4,4,6,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,6,10}*240
   3-fold quotients : {4,2,10}*160
   5-fold quotients : {4,6,2}*96a
   6-fold quotients : {4,2,5}*80, {2,2,10}*80
   10-fold quotients : {2,6,2}*48
   12-fold quotients : {2,2,5}*40
   15-fold quotients : {4,2,2}*32
   20-fold quotients : {2,3,2}*24
   30-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12,10}*960a, {4,6,20}*960a, {8,6,10}*960
   3-fold covers : {4,18,10}*1440a, {12,6,10}*1440a, {12,6,10}*1440c, {4,6,30}*1440a, {4,6,30}*1440b
   4-fold covers : {4,12,20}*1920a, {8,12,10}*1920a, {4,24,10}*1920a, {8,12,10}*1920b, {4,24,10}*1920b, {4,12,10}*1920a, {8,6,20}*1920, {4,6,40}*1920a, {16,6,10}*1920, {4,6,10}*1920b, {4,6,20}*1920b
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)( 2,32)( 3,33)( 4,34)( 5,35)( 6,41)( 7,42)( 8,43)( 9,44)(10,45)
(11,36)(12,37)(13,38)(14,39)(15,40)(16,46)(17,47)(18,48)(19,49)(20,50)(21,56)
(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55);;
s2 := ( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)(16,21)(17,25)(18,24)
(19,23)(20,22)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)
(46,51)(47,55)(48,54)(49,53)(50,52)(57,60)(58,59);;
s3 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)
(26,27)(28,30)(31,32)(33,35)(36,37)(38,40)(41,42)(43,45)(46,47)(48,50)(51,52)
(53,55)(56,57)(58,60);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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(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)( 2,32)( 3,33)( 4,34)( 5,35)( 6,41)( 7,42)( 8,43)( 9,44)
(10,45)(11,36)(12,37)(13,38)(14,39)(15,40)(16,46)(17,47)(18,48)(19,49)(20,50)
(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55);
s2 := Sym(60)!( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)(16,21)(17,25)
(18,24)(19,23)(20,22)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)
(43,44)(46,51)(47,55)(48,54)(49,53)(50,52)(57,60)(58,59);
s3 := Sym(60)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)
(23,25)(26,27)(28,30)(31,32)(33,35)(36,37)(38,40)(41,42)(43,45)(46,47)(48,50)
(51,52)(53,55)(56,57)(58,60);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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