Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1728,47847)
Rank
5
Schläfli Type
{6,4,4,3}
Vertices, edges, …
9, 18, 24, 12, 6
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=<(s2*s3)^2> of order 2

4 facets

9 vertex figures

Representations

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

References

None.

to this polytope.