Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,10}

Atlas Canonical Name {4,4,10}*640

Overview

Group
SmallGroup(640,14119)
Rank
4
Schläfli Type
{4,4,10}
Vertices, edges, …
8, 16, 40, 10
Order of s0s1s2s3
20
Order of s0s1s2s3s2s1
2
Also known as
{{4,4}4,{4,10|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

5-fold

8-fold

10-fold

16-fold

20-fold

40-fold

Covers minimal covers in bold

2-fold

3-fold

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 2

10 facets

  • 10 of 2-fold non-regular quotient of {4,4}*64

4 vertex figures

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

10 facets

  • 10 of 2-fold non-regular quotient of {4,4}*64

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40);;
s1 := (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40);;
s2 := ( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)(11,31)(12,35)(13,34)(14,33)(15,32)(16,36)(17,40)(18,39)(19,38)(20,37);;
s3 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)(26,27)(28,30)(31,32)(33,35)(36,37)(38,40);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(40)!(11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40);
s1 := Sym(40)!(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40);
s2 := Sym(40)!( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)(11,31)(12,35)(13,34)(14,33)(15,32)(16,36)(17,40)(18,39)(19,38)(20,37);
s3 := Sym(40)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)(26,27)(28,30)(31,32)(33,35)(36,37)(38,40);
poly := sub<Sym(40)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.