Polytope of Type {10,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,6}*480
Also Known As : {{10,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(480,1097)
Rank : 4
Schlafli Type : {10,4,6}
Number of vertices, edges, etc : 10, 20, 12, 6
Order of s₀s₁s₂s₃ : 60
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,4,6,2} of size 960
   {10,4,6,3} of size 1440
   {10,4,6,4} of size 1920
   {10,4,6,3} of size 1920
   {10,4,6,4} of size 1920
Vertex Figure Of :
   {2,10,4,6} of size 960
   {4,10,4,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,2,6}*240
   3-fold quotients : {10,4,2}*160
   4-fold quotients : {5,2,6}*120, {10,2,3}*120
   5-fold quotients : {2,4,6}*96a
   6-fold quotients : {10,2,2}*80
   8-fold quotients : {5,2,3}*60
   10-fold quotients : {2,2,6}*48
   12-fold quotients : {5,2,2}*40
   15-fold quotients : {2,4,2}*32
   20-fold quotients : {2,2,3}*24
   30-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {10,4,12}*960, {20,4,6}*960, {10,8,6}*960
   3-fold covers : {10,4,18}*1440, {10,12,6}*1440a, {10,12,6}*1440c, {30,4,6}*1440
   4-fold covers : {20,4,12}*1920, {10,8,12}*1920a, {20,8,6}*1920a, {10,4,24}*1920a, {40,4,6}*1920a, {10,8,12}*1920b, {20,8,6}*1920b, {10,4,24}*1920b, {40,4,6}*1920b, {10,4,12}*1920a, {20,4,6}*1920a, {10,16,6}*1920, {10,4,6}*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);;
s1 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)
(26,27)(28,30)(31,47)(32,46)(33,50)(34,49)(35,48)(36,52)(37,51)(38,55)(39,54)
(40,53)(41,57)(42,56)(43,60)(44,59)(45,58);;
s2 := ( 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);;
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*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*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(60)!( 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);
s1 := 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)(31,47)(32,46)(33,50)(34,49)(35,48)(36,52)(37,51)(38,55)
(39,54)(40,53)(41,57)(42,56)(43,60)(44,59)(45,58);
s2 := 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);
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*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*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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