Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,6,3}

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

Overview

Group
SmallGroup(1944,3579)
Rank
5
Schläfli Type
{3,6,6,3}
Vertices, edges, …
3, 27, 54, 27, 3
Order of s0s1s2s3s4
6
Order of s0s1s2s3s4s3s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat
  • Self-Dual

Quotients maximal quotients in bold

3-fold

9-fold

27-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=<(s2*s3)^2> of order 3

3 facets

3 vertex figures

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

3 facets

3 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16)(20,21)(22,23)(25,27);;
s2 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);;
s3 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,14)(11,13)(12,15)(16,17)(19,24)(20,23)(21,22)(25,27);;
s4 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s4*s2*s3*s2*s3*s4*s2*s3, s3*s0*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(27)!( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26);
s1 := Sym(27)!( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16)(20,21)(22,23)(25,27);
s2 := Sym(27)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);
s3 := Sym(27)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,14)(11,13)(12,15)(16,17)(19,24)(20,23)(21,22)(25,27);
s4 := 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);
poly := sub<Sym(27)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s4*s2*s3*s2*s3*s4*s2*s3, 
s3*s0*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s0*s1*s2*s1 >;

References

None.

to this polytope.