Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3,6}

Atlas Canonical Name {6,3,6}*1944a

Overview

Group
SmallGroup(1944,2342)
Rank
4
Schläfli Type
{6,3,6}
Vertices, edges, …
18, 81, 81, 18
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Locally Toroidal
  • Orientable
  • Flat
  • Self-Dual

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

81-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^2> of order 3

18 facets

  • 18 of 3-fold non-regular quotient of {6,3}*108

6 vertex figures

P/N, where N=<s1*(s2*s1*s3)^2*s2> of order 3

6 facets

18 vertex figures

  • 18 of 3-fold non-regular quotient of {3,6}*108
P/N, where N=<s0*s2*s1*s0*s1*s2, s1*(s2*s1*s3)^2*s2> of order 9

6 facets

  • 6 of 3-fold non-regular quotient of {6,3}*108

6 vertex figures

  • 6 of 3-fold non-regular quotient of {3,6}*108
P/N, where N=<(s0*s1)^2, s1*(s2*s1*s3)^2*s2> of order 9

6 facets

  • 6 of 3-fold non-regular quotient of {6,3}*108

6 vertex figures

  • 6 of 3-fold non-regular quotient of {3,6}*108

Representations

Permutation Representation (GAP) 
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);;
s1 := (10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24);;
s2 := ( 1,11)( 2,12)( 3,10)( 4,14)( 5,15)( 6,13)( 7,17)( 8,18)( 9,16);;
s3 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27);;
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, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(27)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);
s1 := Sym(27)!(10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24);
s2 := Sym(27)!( 1,11)( 2,12)( 3,10)( 4,14)( 5,15)( 6,13)( 7,17)( 8,18)( 9,16);
s3 := Sym(27)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27);
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, 
s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;

References

None.

to this polytope.