Polytope of Type {10,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6,3}*360
if this polytope has a name.
Group : SmallGroup(360,137)
Rank : 4
Schlafli Type : {10,6,3}
Number of vertices, edges, etc : 10, 30, 9, 3
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,6,3,2} of size 720
   {10,6,3,4} of size 1440
Vertex Figure Of :
   {2,10,6,3} of size 720
   {4,10,6,3} of size 1440
   {5,10,6,3} of size 1800
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {10,2,3}*120
   5-fold quotients : {2,6,3}*72
   6-fold quotients : {5,2,3}*60
   15-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,6,3}*720, {10,6,6}*720b
   3-fold covers : {10,6,9}*1080, {10,6,3}*1080, {30,6,3}*1080b
   4-fold covers : {40,6,3}*1440, {10,6,12}*1440b, {20,6,6}*1440c, {10,12,6}*1440c, {10,6,3}*1440, {10,12,3}*1440
   5-fold covers : {50,6,3}*1800, {10,6,15}*1800
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);;
s1 := ( 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);;
s2 := ( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(16,36)(17,37)(18,38)(19,39)(20,40)
(21,31)(22,32)(23,33)(24,34)(25,35)(26,41)(27,42)(28,43)(29,44)(30,45);;
s3 := ( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)
(11,21)(12,22)(13,23)(14,24)(15,25)(36,41)(37,42)(38,43)(39,44)(40,45);;
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, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
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(45)!( 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);
s1 := Sym(45)!( 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);
s2 := Sym(45)!( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(16,36)(17,37)(18,38)(19,39)
(20,40)(21,31)(22,32)(23,33)(24,34)(25,35)(26,41)(27,42)(28,43)(29,44)(30,45);
s3 := Sym(45)!( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)( 6,26)( 7,27)( 8,28)( 9,29)
(10,30)(11,21)(12,22)(13,23)(14,24)(15,25)(36,41)(37,42)(38,43)(39,44)(40,45);
poly := sub<Sym(45)|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, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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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