Polytope of Type {3,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6}*1944d
if this polytope has a name.
Group : SmallGroup(1944,2342)
Rank : 4
Schlafli Type : {3,6,6}
Number of vertices, edges, etc : 3, 81, 162, 54
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6,6}*648d, {3,6,6}*648e
   9-fold quotients : {3,6,6}*216a, {3,6,6}*216b
   18-fold quotients : {3,6,3}*108
   27-fold quotients : {3,2,6}*72, {3,6,2}*72
   54-fold quotients : {3,2,3}*36
   81-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s2*s3*s2*s3*s2*s3> of order 2.
      27 facets:
         27 of {3,6}*36
      3 vertex figures:
         3 of 2-fold non-regular quotient of {6,6}*648g
   P/N, where N=<s1*s2*s3*s2*s1*s2*s3*s2> of order 3.
      18 facets:
         18 of {3,6}*36
      3 vertex figures:
         3 of 3-fold non-regular quotient of {6,6}*648g
   P/N, where N=<s1*s2*s1*s2*s1*s3*s2*s1*s2*s3> of order 3.
      18 facets:
         18 of {3,6}*36
      3 vertex figures:
         3 of 3-fold non-regular quotient of {6,6}*648g
   P/N, where N=<s2*s1*s3*s2*s1*s2*s3*s2> of order 3.
      24 facets:
         15 of {3,6}*36
         9 of {3,2}*12
      3 vertex figures:
         3 of 3-fold non-regular quotient of {6,6}*648g
   P/N, where N=<s2*s3*s2*s3> of order 3.
      18 facets:
         18 of {3,6}*36
      3 vertex figures:
         2 of 3-fold non-regular quotient of {6,6}*648g
         1 of 3-fold non-regular quotient of {6,6}*648g
   P/N, where N=<s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s3*s2*s1*s3> of order 6.
      9 facets:
         9 of {3,6}*36
      3 vertex figures:
         3 of 6-fold non-regular quotient of {6,6}*648g
   P/N, where N=<s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s3*s2*s1*s2*s3> of order 9.
      10 facets:
         4 of {3,6}*36
         6 of {3,2}*12
      3 vertex figures:
         3 of 9-fold non-regular quotient of {6,6}*648g
   P/N, where N=<s2*s3*s2*s3, s1*s2*s3*s2*s1*s3> of order 9.
      6 facets:
         6 of {3,6}*36
      3 vertex figures:
         3 of 9-fold non-regular quotient of {6,6}*648g

Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);;
s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,14)(11,13)(12,15)(16,17)(19,27)(20,26)(21,25)(22,24);;
s2 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s3*s0*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);
s1 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,14)(11,13)(12,15)(16,17)(19,27)(20,26)(21,25)(22,24);
s2 := Sym(27)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27);
s3 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23);
poly := sub<Sym(27)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s3*s0*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s0 >;

References : None.
to this polytope