Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*216d
if this polytope has a name.
Group : SmallGroup(216,162)
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
   Self-Petrie
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
   {6,6,4} 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
   {6,6,6} 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
   {4,6,6} 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
   {6,6,6} of size 1944
   {3,6,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   6-fold quotients : {3,6}*36, {6,3}*36
   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}*432g, {12,6}*432g
   3-fold covers : {6,18}*648i, {18,6}*648i, {6,6}*648e, {6,6}*648f, {6,6}*648g
   4-fold covers : {6,24}*864f, {24,6}*864f, {12,12}*864h, {6,6}*864c, {6,12}*864c, {12,6}*864c
   5-fold covers : {6,30}*1080d, {30,6}*1080d
   6-fold covers : {6,36}*1296l, {36,6}*1296l, {12,18}*1296l, {18,12}*1296l, {6,12}*1296g, {6,12}*1296h, {12,6}*1296g, {12,6}*1296h, {6,12}*1296i, {12,6}*1296i
   7-fold covers : {6,42}*1512d, {42,6}*1512d
   8-fold covers : {6,48}*1728f, {48,6}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {6,12}*1728g, {6,24}*1728f, {12,6}*1728g, {24,6}*1728f, {6,6}*1728f, {6,24}*1728g, {24,6}*1728g, {12,12}*1728v, {12,12}*1728w, {6,12}*1728h, {6,12}*1728i, {12,6}*1728h, {12,6}*1728i, {12,12}*1728x, {12,12}*1728y
   9-fold covers : {18,18}*1944ad, {18,18}*1944ae, {18,18}*1944af, {6,18}*1944m, {6,18}*1944n, {18,6}*1944m, {18,6}*1944n, {6,18}*1944o, {18,6}*1944o, {6,6}*1944d, {6,6}*1944e, {6,6}*1944f, {6,54}*1944g, {54,6}*1944g, {6,6}*1944g, {6,6}*1944h, {6,18}*1944p, {6,18}*1944q, {18,6}*1944p, {18,6}*1944q, {6,18}*1944r, {6,18}*1944s, {18,6}*1944r, {18,6}*1944s, {6,6}*1944i, {6,6}*1944j, {6,18}*1944t, {6,18}*1944u, {18,6}*1944t, {18,6}*1944u
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)
(17,23)(18,24);;
s1 := ( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)
(20,24)(21,23)(26,27);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)
(17,25)(18,27);;
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*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(27)!( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)
(16,22)(17,23)(18,24);
s1 := Sym(27)!( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)
(19,22)(20,24)(21,23)(26,27);
s2 := Sym(27)!( 1, 2)( 4, 5)( 7, 8)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)
(16,26)(17,25)(18,27);
poly := sub<Sym(27)|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*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 >;

References : None.
to this polytope