Polytope of Type {12,15}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,15}*720
if this polytope has a name.
Group : SmallGroup(720,793)
Rank : 3
Schlafli Type : {12,15}
Number of vertices, edges, etc : 24, 180, 30
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,15,2} of size 1440
Vertex Figure Of :
   {2,12,15} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,15}*240
   4-fold quotients : {6,15}*180
   5-fold quotients : {12,3}*144
   6-fold quotients : {4,15}*120
   12-fold quotients : {2,15}*60
   15-fold quotients : {4,3}*48
   20-fold quotients : {6,3}*36
   30-fold quotients : {4,3}*24
   36-fold quotients : {2,5}*20
   60-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,15}*1440, {12,30}*1440b
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

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