Polytope of Type {3,2,15}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,15}*180
if this polytope has a name.
Group : SmallGroup(180,29)
Rank : 4
Schlafli Type : {3,2,15}
Number of vertices, edges, etc : 3, 3, 15, 15
Order of s₀s₁s₂s₃ : 15
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,15,2} of size 360
   {3,2,15,4} of size 720
   {3,2,15,6} of size 1080
   {3,2,15,6} of size 1440
   {3,2,15,4} of size 1440
   {3,2,15,10} of size 1800
Vertex Figure Of :
   {2,3,2,15} of size 360
   {3,3,2,15} of size 720
   {4,3,2,15} of size 720
   {6,3,2,15} of size 1080
   {4,3,2,15} of size 1440
   {6,3,2,15} of size 1440
   {5,3,2,15} of size 1800
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,5}*60
   5-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,30}*360, {6,2,15}*360
   3-fold covers : {3,2,45}*540, {9,2,15}*540, {3,6,15}*540
   4-fold covers : {12,2,15}*720, {3,2,60}*720, {6,2,30}*720
   5-fold covers : {3,2,75}*900, {15,2,15}*900
   6-fold covers : {3,2,90}*1080, {6,2,45}*1080, {9,2,30}*1080, {18,2,15}*1080, {3,6,30}*1080a, {6,6,15}*1080a, {3,6,30}*1080b, {6,6,15}*1080b
   7-fold covers : {21,2,15}*1260, {3,2,105}*1260
   8-fold covers : {24,2,15}*1440, {3,2,120}*1440, {12,2,30}*1440, {6,2,60}*1440, {6,4,30}*1440, {6,4,15}*1440, {3,4,30}*1440
   9-fold covers : {9,2,45}*1620, {3,6,45}*1620, {9,6,15}*1620, {3,2,135}*1620, {27,2,15}*1620, {3,6,15}*1620a, {3,6,15}*1620b
   10-fold covers : {3,2,150}*1800, {6,2,75}*1800, {6,10,15}*1800, {15,2,30}*1800, {30,2,15}*1800
   11-fold covers : {33,2,15}*1980, {3,2,165}*1980
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);;
s3 := ( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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(18)!(2,3);
s1 := Sym(18)!(1,2);
s2 := Sym(18)!( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18);
s3 := Sym(18)!( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17);
poly := sub<Sym(18)|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, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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