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Polytope of Type {3,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,12}*144
if this polytope has a name.
Group : SmallGroup(144,183)
Rank : 3
Schlafli Type : {3,12}
Number of vertices, edges, etc : 6, 36, 24
Order of s0s1s2 : 6
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {3,12,2} of size 288
   {3,12,4} of size 576
   {3,12,6} of size 864
   {3,12,6} of size 864
   {3,12,4} of size 1152
   {3,12,8} of size 1152
   {3,12,10} of size 1440
   {3,12,12} of size 1728
   {3,12,3} of size 1728
   {3,12,12} of size 1728
   {3,12,4} of size 1728
Vertex Figure Of :
   {2,3,12} of size 288
   {4,3,12} of size 576
   {6,3,12} of size 864
   {4,3,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,4}*48
   4-fold quotients : {3,6}*36
   6-fold quotients : {3,4}*24
   12-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,24}*288, {6,12}*288b
   3-fold covers : {9,12}*432, {3,12}*432
   4-fold covers : {3,24}*576, {12,12}*576e, {12,12}*576h, {6,24}*576b, {6,24}*576d, {6,12}*576f, {3,12}*576
   5-fold covers : {15,12}*720
   6-fold covers : {9,24}*864, {3,24}*864, {18,12}*864b, {6,12}*864a, {6,12}*864c
   7-fold covers : {21,12}*1008
   8-fold covers : {3,24}*1152a, {6,24}*1152a, {6,24}*1152b, {12,24}*1152j, {12,24}*1152l, {12,24}*1152m, {12,24}*1152n, {6,12}*1152c, {6,24}*1152f, {24,12}*1152p, {24,12}*1152r, {24,12}*1152s, {24,12}*1152t, {12,12}*1152m, {6,24}*1152k, {6,24}*1152l, {12,12}*1152s, {6,12}*1152f, {3,12}*1152b, {3,24}*1152b, {6,12}*1152g, {3,24}*1152c, {6,12}*1152j
   9-fold covers : {27,12}*1296, {9,36}*1296, {3,36}*1296, {3,12}*1296a, {9,12}*1296a, {9,12}*1296b, {9,12}*1296c, {9,12}*1296d
   10-fold covers : {15,24}*1440, {6,60}*1440c, {30,12}*1440b
   11-fold covers : {33,12}*1584
   12-fold covers : {9,24}*1728, {3,24}*1728, {36,12}*1728f, {36,12}*1728g, {12,12}*1728i, {12,12}*1728m, {18,24}*1728b, {18,24}*1728d, {6,24}*1728b, {6,24}*1728d, {18,12}*1728d, {6,12}*1728f, {9,12}*1728, {3,12}*1728, {6,24}*1728f, {6,24}*1728g, {12,12}*1728v, {6,12}*1728i, {12,12}*1728x
   13-fold covers : {39,12}*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)( 3, 4)( 5,10)( 6, 9)( 7,12)( 8,11);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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)( 3, 4)( 5,10)( 6, 9)( 7,12)( 8,11);
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 >; 
 
References : None.
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