Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,4}

Atlas Canonical Name {4,6,4}*1920a

Overview

Group
SmallGroup(1920,240594)
Rank
4
Schläfli Type
{4,6,4}
Vertices, edges, …
4, 120, 120, 40
Order of s0s1s2s3
20
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

60-fold

120-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*s1*s3)^4*s2*s3> of order 2

20 facets

4 vertex figures

  • 4 of 2-fold non-regular quotient of {6,4}*480
P/N, where N=<(s1*s2)^2*(s3*s2*s1*s2)^2> of order 2

20 facets

4 vertex figures

  • 4 of 2-fold non-regular quotient of {6,4}*480
P/N, where N=<(s1*s2)^2> of order 3

16 facets

4 vertex figures

  • 4 of 3-fold non-regular quotient of {6,4}*480
P/N, where N=<(s1*s2)^2*s1*s3*s2*s1*s2*s3, s1*s3*(s2*s1)^2*s3*s2*s1*s2> of order 4

10 facets

4 vertex figures

  • 4 of 4-fold non-regular quotient of {6,4}*480
P/N, where N=<(s1*s2)^2*(s3*s2*s1*s2)^2, (s2*s1*s3)^4*s2*s3> of order 4

10 facets

4 vertex figures

  • 4 of 4-fold non-regular quotient of {6,4}*480
P/N, where N=<(s1*s2)^2, s1*s3*s2*s1*s3*(s2*s1)^2*s3*s2*s3> of order 6

8 facets

4 vertex figures

  • 4 of 6-fold non-regular quotient of {6,4}*480
P/N, where N=<(s1*s2)^2, s1*s2*(s3*s2*s1)^3*s3*s2*s3> of order 6

8 facets

4 vertex figures

  • 4 of 6-fold non-regular quotient of {6,4}*480
P/N, where N=<(s1*s2)^2, s1*s3*s2*s1*s2*s3> of order 12

6 facets

4 vertex figures

  • 4 of 12-fold non-regular quotient of {6,4}*480

Representations

Permutation Representation (GAP) 
s0 := ( 6,16)( 7,17)( 8,14)( 9,15)(10,20)(11,21)(12,18)(13,19)(22,32)(23,33)(24,30)(25,31)(26,36)(27,37)(28,34)(29,35);;
s1 := ( 4, 5)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)(11,27)(12,28)(13,29)(14,31)(15,30)(16,33)(17,32)(18,35)(19,34)(20,37)(21,36);;
s2 := ( 1, 2)( 3, 4)( 6,12)( 7,13)( 8,10)( 9,11)(14,20)(15,21)(16,18)(17,19)(22,28)(23,29)(24,26)(25,27)(30,36)(31,37)(32,34)(33,35);;
s3 := ( 2, 3)( 6, 8)( 7, 9)(10,12)(11,13)(14,16)(15,17)(18,20)(19,21)(22,24)(23,25)(26,28)(27,29)(30,32)(31,33)(34,36)(35,37);;
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*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(37)!( 6,16)( 7,17)( 8,14)( 9,15)(10,20)(11,21)(12,18)(13,19)(22,32)(23,33)(24,30)(25,31)(26,36)(27,37)(28,34)(29,35);
s1 := Sym(37)!( 4, 5)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)(11,27)(12,28)(13,29)(14,31)(15,30)(16,33)(17,32)(18,35)(19,34)(20,37)(21,36);
s2 := Sym(37)!( 1, 2)( 3, 4)( 6,12)( 7,13)( 8,10)( 9,11)(14,20)(15,21)(16,18)(17,19)(22,28)(23,29)(24,26)(25,27)(30,36)(31,37)(32,34)(33,35);
s3 := Sym(37)!( 2, 3)( 6, 8)( 7, 9)(10,12)(11,13)(14,16)(15,17)(18,20)(19,21)(22,24)(23,25)(26,28)(27,29)(30,32)(31,33)(34,36)(35,37);
poly := sub<Sym(37)|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*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2 >;

References

None.

to this polytope.