Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*108
Also Known As : {6,6}₃. if this polytope has another name.
Group : SmallGroup(108,17)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 9, 27, 9
Order of s₀s₁s₂ : 3
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,6,2} of size 216
   {6,6,4} of size 432
   {6,6,6} of size 648
   {6,6,4} of size 864
   {6,6,8} of size 1728
   {6,6,6} of size 1944
   {6,6,6} of size 1944
Vertex Figure Of :
   {2,6,6} of size 216
   {4,6,6} of size 432
   {6,6,6} of size 648
   {4,6,6} of size 864
   {8,6,6} of size 1728
   {6,6,6} of size 1944
   {6,6,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6}*216b
   3-fold covers : {6,18}*324a, {18,6}*324a, {6,6}*324a, {6,6}*324b, {6,18}*324b, {18,6}*324b, {6,18}*324c, {18,6}*324c
   4-fold covers : {6,12}*432b, {12,6}*432b, {6,12}*432d, {12,6}*432d
   5-fold covers : {6,30}*540, {30,6}*540
   6-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
   7-fold covers : {6,42}*756, {42,6}*756
   8-fold covers : {6,24}*864b, {24,6}*864b, {12,12}*864c, {6,12}*864b, {12,6}*864b
   9-fold covers : {18,18}*972a, {18,18}*972b, {6,6}*972, {6,18}*972a, {18,6}*972a, {6,18}*972b, {18,6}*972b, {18,18}*972c, {18,18}*972d, {18,18}*972e, {6,54}*972a, {54,6}*972a, {6,18}*972c, {18,6}*972c, {18,18}*972f, {18,18}*972g, {18,18}*972h, {18,18}*972i, {6,18}*972d, {18,6}*972d, {6,54}*972b, {54,6}*972b, {6,54}*972c, {54,6}*972c, {6,18}*972e, {18,6}*972e
   10-fold covers : {6,30}*1080c, {30,6}*1080c
   11-fold covers : {6,66}*1188, {66,6}*1188
   12-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, {6,36}*1296i, {36,6}*1296i, {6,36}*1296j, {36,6}*1296j, {6,36}*1296k, {36,6}*1296k, {12,18}*1296i, {18,12}*1296i, {12,18}*1296j, {18,12}*1296j, {6,12}*1296e, {12,6}*1296e, {12,18}*1296k, {18,12}*1296k, {6,12}*1296f, {12,6}*1296f, {6,12}*1296g, {12,6}*1296g
   13-fold covers : {6,78}*1404, {78,6}*1404
   14-fold covers : {6,42}*1512c, {42,6}*1512c
   15-fold covers : {6,90}*1620a, {90,6}*1620a, {18,30}*1620a, {30,18}*1620a, {6,30}*1620a, {30,6}*1620a, {6,30}*1620b, {30,6}*1620b, {6,90}*1620b, {90,6}*1620b, {18,30}*1620b, {30,18}*1620b, {6,90}*1620c, {90,6}*1620c, {18,30}*1620c, {30,18}*1620c
   16-fold covers : {6,48}*1728b, {48,6}*1728b, {12,12}*1728c, {12,24}*1728d, {24,12}*1728d, {12,24}*1728f, {24,12}*1728f, {6,24}*1728a, {24,6}*1728a, {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, {12,12}*1728u, {6,6}*1728d
   17-fold covers : {6,102}*1836, {102,6}*1836
   18-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 := (4,5)(6,7)(8,9);;
s1 := (2,6)(3,4)(5,7);;
s2 := (1,2)(4,9)(5,8)(6,7);;
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*s2*s0*s1*s2*s0*s1*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope