Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*144
Also Known As : {3,6}_(2,2). if this polytope has another name.
Group : SmallGroup(144,183)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 12, 36, 24
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {3,6,2} of size 288
   {3,6,4} of size 576
   {3,6,3} of size 720
   {3,6,6} of size 864
   {3,6,4} of size 1152
   {3,6,8} of size 1152
   {3,6,4} of size 1152
   {3,6,6} of size 1440
   {3,6,10} of size 1440
   {3,6,12} of size 1728
Vertex Figure Of :
   {2,3,6} of size 288
   {4,3,6} of size 576
   {3,3,6} of size 720
   {6,3,6} of size 864
   {4,3,6} of size 1152
   {4,3,6} of size 1152
   {6,3,6} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*48
   4-fold quotients : {3,6}*36
   6-fold quotients : {3,3}*24
   12-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,12}*288, {6,6}*288b
   3-fold covers : {9,6}*432, {3,6}*432
   4-fold covers : {3,6}*576, {12,6}*576a, {6,12}*576c, {6,6}*576b, {12,6}*576d, {6,12}*576e, {3,12}*576
   5-fold covers : {15,6}*720e
   6-fold covers : {9,12}*864, {3,12}*864, {18,6}*864, {6,6}*864a, {6,6}*864c
   7-fold covers : {21,6}*1008b
   8-fold covers : {3,12}*1152a, {6,6}*1152a, {12,12}*1152e, {12,12}*1152g, {12,6}*1152a, {6,6}*1152d, {6,6}*1152f, {24,6}*1152g, {24,6}*1152i, {12,12}*1152l, {6,24}*1152j, {6,12}*1152e, {12,12}*1152q, {6,24}*1152m, {3,12}*1152b, {3,24}*1152b, {6,12}*1152g, {3,24}*1152c, {6,12}*1152j
   9-fold covers : {27,6}*1296, {9,18}*1296a, {9,6}*1296a, {3,6}*1296, {9,6}*1296b, {3,18}*1296a, {9,6}*1296c, {9,6}*1296d
   10-fold covers : {15,12}*1440c, {6,30}*1440g, {30,6}*1440h
   11-fold covers : {33,6}*1584
   12-fold covers : {9,6}*1728, {3,6}*1728, {36,6}*1728a, {18,12}*1728a, {18,6}*1728a, {36,6}*1728c, {18,12}*1728b, {12,6}*1728a, {6,12}*1728c, {6,6}*1728b, {12,6}*1728d, {6,12}*1728e, {9,12}*1728, {3,12}*1728, {6,12}*1728g, {12,6}*1728g, {6,6}*1728f, {6,12}*1728h, {12,6}*1728h
   13-fold covers : {39,6}*1872
Permutation Representation (GAP) :

s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11);;
s1 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11);;
s2 := ( 1, 2)( 5, 6)( 9,10);;
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*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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