Polytope of Type {15,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {15,6}*180
if this polytope has a name.
Group : SmallGroup(180,29)
Rank : 3
Schlafli Type : {15,6}
Number of vertices, edges, etc : 15, 45, 6
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {15,6,2} of size 360
   {15,6,3} of size 540
   {15,6,4} of size 720
   {15,6,6} of size 1080
   {15,6,6} of size 1080
   {15,6,8} of size 1440
   {15,6,9} of size 1620
   {15,6,3} of size 1620
   {15,6,10} of size 1800
Vertex Figure Of :
   {2,15,6} of size 360
   {4,15,6} of size 720
   {6,15,6} of size 1080
   {4,15,6} of size 1440
   {10,15,6} of size 1800
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {15,2}*60
   5-fold quotients : {3,6}*36
   9-fold quotients : {5,2}*20
   15-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {30,6}*360c
   3-fold covers : {45,6}*540, {15,6}*540
   4-fold covers : {60,6}*720c, {30,12}*720c, {15,12}*720, {15,6}*720e
   5-fold covers : {75,6}*900, {15,30}*900
   6-fold covers : {90,6}*1080b, {30,6}*1080b, {30,6}*1080d
   7-fold covers : {105,6}*1260
   8-fold covers : {120,6}*1440c, {60,12}*1440c, {30,24}*1440c, {15,24}*1440, {15,12}*1440c, {30,12}*1440b, {30,6}*1440h
   9-fold covers : {45,18}*1620, {45,6}*1620a, {135,6}*1620, {45,6}*1620b, {45,6}*1620c, {45,6}*1620d, {15,6}*1620, {15,18}*1620
   10-fold covers : {150,6}*1800c, {30,30}*1800a, {30,30}*1800i
   11-fold covers : {165,6}*1980
Permutation Representation (GAP) :

s0 := ( 2, 5)( 3, 4)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(16,31)(17,35)(18,34)
(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)(29,38)
(30,37);;
s1 := ( 1,22)( 2,21)( 3,25)( 4,24)( 5,23)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)
(11,27)(12,26)(13,30)(14,29)(15,28)(31,37)(32,36)(33,40)(34,39)(35,38)(41,42)
(43,45);;
s2 := (16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)
(26,41)(27,42)(28,43)(29,44)(30,45);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(16,31)(17,35)
(18,34)(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)
(29,38)(30,37);
s1 := Sym(45)!( 1,22)( 2,21)( 3,25)( 4,24)( 5,23)( 6,17)( 7,16)( 8,20)( 9,19)
(10,18)(11,27)(12,26)(13,30)(14,29)(15,28)(31,37)(32,36)(33,40)(34,39)(35,38)
(41,42)(43,45);
s2 := Sym(45)!(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)
(25,40)(26,41)(27,42)(28,43)(29,44)(30,45);
poly := sub<Sym(45)|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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope