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Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*216b
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 18, 54, 18
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,6,2} of size 432
   {6,6,3} of size 648
   {6,6,4} of size 864
   {6,6,4} of size 864
   {6,6,6} of size 1296
   {6,6,6} of size 1296
   {6,6,6} of size 1296
   {6,6,8} of size 1728
   {6,6,4} of size 1728
   {6,6,9} of size 1944
   {6,6,3} of size 1944
Vertex Figure Of :
   {2,6,6} of size 432
   {3,6,6} of size 648
   {4,6,6} of size 864
   {4,6,6} of size 864
   {6,6,6} of size 1296
   {6,6,6} of size 1296
   {6,6,6} of size 1296
   {8,6,6} of size 1728
   {4,6,6} of size 1728
   {9,6,6} of size 1944
   {3,6,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6}*108
   3-fold quotients : {6,6}*72a
   9-fold quotients : {2,6}*24, {6,2}*24
   18-fold quotients : {2,3}*12, {3,2}*12
   27-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,12}*432b, {12,6}*432b
   3-fold covers : {6,18}*648b, {18,6}*648b, {6,6}*648a, {6,6}*648b, {6,18}*648f, {18,6}*648f, {6,18}*648g, {18,6}*648g, {6,6}*648g
   4-fold covers : {6,24}*864b, {24,6}*864b, {12,12}*864c, {6,12}*864b, {12,6}*864b
   5-fold covers : {6,30}*1080c, {30,6}*1080c
   6-fold covers : {12,18}*1296a, {18,12}*1296a, {6,36}*1296b, {36,6}*1296b, {6,12}*1296a, {12,6}*1296a, {6,12}*1296b, {12,6}*1296b, {12,18}*1296b, {18,12}*1296b, {6,36}*1296f, {36,6}*1296f, {12,18}*1296c, {18,12}*1296c, {6,36}*1296g, {36,6}*1296g, {6,12}*1296g, {12,6}*1296g
   7-fold covers : {6,42}*1512c, {42,6}*1512c
   8-fold covers : {6,48}*1728b, {48,6}*1728b, {12,12}*1728c, {12,24}*1728d, {24,12}*1728d, {12,24}*1728f, {24,12}*1728f, {12,12}*1728j, {12,12}*1728l, {6,12}*1728b, {12,6}*1728b, {6,24}*1728c, {24,6}*1728c, {6,24}*1728e, {24,6}*1728e, {12,12}*1728o, {12,12}*1728p
   9-fold covers : {18,18}*1944c, {6,6}*1944a, {18,18}*1944d, {6,18}*1944c, {18,6}*1944c, {6,18}*1944e, {18,6}*1944e, {18,18}*1944i, {18,18}*1944k, {18,18}*1944m, {6,54}*1944b, {54,6}*1944b, {6,18}*1944g, {18,6}*1944g, {18,18}*1944s, {18,18}*1944v, {18,18}*1944x, {18,18}*1944z, {6,18}*1944j, {18,6}*1944j, {6,54}*1944d, {54,6}*1944d, {6,54}*1944f, {54,6}*1944f, {6,18}*1944l, {18,6}*1944l, {6,18}*1944n, {18,6}*1944n, {6,6}*1944e, {6,6}*1944f, {6,6}*1944g, {6,6}*1944h, {6,18}*1944s, {18,6}*1944s, {6,18}*1944t, {18,6}*1944t
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);;
s1 := ( 4, 8)( 5, 9)( 6, 7)(13,17)(14,18)(15,16);;
s2 := ( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(18)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);
s1 := Sym(18)!( 4, 8)( 5, 9)( 6, 7)(13,17)(14,18)(15,16);
s2 := Sym(18)!( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1 >; 
 
References : None.
to this polytope