Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,4,4,6}

Atlas Canonical Name {3,4,4,6}*1728

Overview

Group
SmallGroup(1728,47847)
Rank
5
Schläfli Type
{3,4,4,6}
Vertices, edges, …
6, 12, 24, 18, 9
Order of s0s1s2s3s4
12
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

4-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*s0*s2)^2*s1> of order 2

9 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)(31,32)(35,36);;
s1 := ( 2, 4)( 6, 8)(10,12)(14,16)(18,20)(22,24)(26,28)(30,32)(34,36);;
s2 := ( 1, 2)( 3, 4)( 5,30)( 6,29)( 7,32)( 8,31)( 9,22)(10,21)(11,24)(12,23)(13,26)(14,25)(15,28)(16,27)(17,18)(19,20)(33,34)(35,36);;
s3 := (13,33)(14,34)(15,35)(16,36)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32);;
s4 := ( 1,17)( 2,18)( 3,19)( 4,20)( 5,13)( 6,14)( 7,15)( 8,16)( 9,21)(10,22)(11,23)(12,24)(25,29)(26,30)(27,31)(28,32);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s3*s4*s3*s2*s3*s4*s2*s3*s2*s3*s4*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(36)!( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)(31,32)(35,36);
s1 := Sym(36)!( 2, 4)( 6, 8)(10,12)(14,16)(18,20)(22,24)(26,28)(30,32)(34,36);
s2 := Sym(36)!( 1, 2)( 3, 4)( 5,30)( 6,29)( 7,32)( 8,31)( 9,22)(10,21)(11,24)(12,23)(13,26)(14,25)(15,28)(16,27)(17,18)(19,20)(33,34)(35,36);
s3 := Sym(36)!(13,33)(14,34)(15,35)(16,36)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32);
s4 := Sym(36)!( 1,17)( 2,18)( 3,19)( 4,20)( 5,13)( 6,14)( 7,15)( 8,16)( 9,21)(10,22)(11,23)(12,24)(25,29)(26,30)(27,31)(28,32);
poly := sub<Sym(36)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s3*s4*s3*s2*s3*s4*s2*s3*s2*s3*s4*s3*s2 >;

References

None.

to this polytope.