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Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*216a
if this polytope has a name.
Group : SmallGroup(216,87)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 18, 54, 9
Order of s0s1s2 : 4
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,6,2} of size 432
Vertex Figure Of :
   {2,12,6} of size 432
   {4,12,6} of size 864
   {6,12,6} of size 1296
   {8,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6}*432f
   3-fold covers : {12,6}*648
   4-fold covers : {24,6}*864e, {12,12}*864e
   5-fold covers : {60,6}*1080b
   6-fold covers : {12,6}*1296j, {12,6}*1296m, {12,6}*1296o
   7-fold covers : {84,6}*1512b
   8-fold covers : {48,6}*1728d, {12,12}*1728d, {24,12}*1728g, {12,24}*1728i, {12,24}*1728k, {24,12}*1728m
   9-fold covers : {12,18}*1944b, {12,6}*1944a, {12,18}*1944d, {12,18}*1944f, {36,6}*1944
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17);;
s1 := ( 1,11)( 2,10)( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18);;
s2 := ( 1, 4)( 2, 5)( 3, 6)(13,18)(14,16)(15,17);;
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*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(18)!( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17);
s1 := Sym(18)!( 1,11)( 2,10)( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18);
s2 := Sym(18)!( 1, 4)( 2, 5)( 3, 6)(13,18)(14,16)(15,17);
poly := sub<Sym(18)|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*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2 >; 
 
References : None.
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