Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,3,5}

Atlas Canonical Name {4,3,5}*1920

Overview

Group
SmallGroup(1920,240995)
Rank
4
Schläfli Type
{4,3,5}
Vertices, edges, …
32, 96, 120, 40
Order of s0s1s2s3
10
Order of s0s1s2s3s2s1
4
Also known as
hemi-2{3,5}/2. if this polytope has another name.

Special Properties

  • Locally Projective
  • Non-Orientable

Quotients maximal quotients in bold

16-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=<s1*s0*s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2*s1> of order 2

22 facets

16 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0*s1*s2*s3)^2> of order 2

24 facets

16 vertex figures

P/N, where N=<(s0*s1)^2, (s2*s1*s0*s1*s2*s3)^2> of order 4

16 facets

8 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s1*s3*s2*s1*s0*s1*s2*s3> of order 4

16 facets

8 vertex figures

P/N, where N=<(s1*s0*s1*s2*s3*s2)^2, s1*s0*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1> of order 4

16 facets

8 vertex figures

P/N, where N=<(s3*s2*s1*s0*s1)^2*(s2*s3)^2, (s0*s1)^2*(s2*s1*s0*s1*s2*s3)^2> of order 4

16 facets

8 vertex figures

P/N, where N=<s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2, s0*(s2*s1*s0*s1*s2*s3)^2> of order 4

14 facets

8 vertex figures

P/N, where N=<s0*(s3*s2*s1*s0*s1)^2*(s2*s3)^2, s1*s0*s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2*s1> of order 4

14 facets

8 vertex figures

P/N, where N=<(s0*s1)^2, (s2*s1*s0*s1*s2*s3)^2, s0*s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2> of order 8

10 facets

4 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s1*s3*s2*s1*s0*s1*s2*s3, (s0*s1)^2*s2*s3*s2*s1*s0*s1*s2*s3*s2> of order 8

10 facets

4 vertex figures

P/N, where N=<(s0*s1)^2, (s2*s1*s0*s1*s2*s3)^2, s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2> of order 8

12 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,20)(18,19);;
s1 := ( 1, 2)( 3, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,18)(14,19)(15,17)(16,20);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20);;
s3 := ( 1, 3)( 2, 4)( 5,17)( 6,20)( 7,18)( 8,19)( 9,15)(10,14)(11,13)(12,16);;
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, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3, 
s0*s2*s1*s0*s2*s3*s2*s1*s2*s0*s3*s1*s2*s3*s1*s0*s1*s2*s3*s1*s0*s1*s2*s3*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(20)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,20)(18,19);
s1 := Sym(20)!( 1, 2)( 3, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,18)(14,19)(15,17)(16,20);
s2 := Sym(20)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20);
s3 := Sym(20)!( 1, 3)( 2, 4)( 5,17)( 6,20)( 7,18)( 8,19)( 9,15)(10,14)(11,13)(12,16);
poly := sub<Sym(20)|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, 
s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3, 
s0*s2*s1*s0*s2*s3*s2*s1*s2*s0*s3*s1*s2*s3*s1*s0*s1*s2*s3*s1*s0*s1*s2*s3*s0*s1 >;

References

  1. Hartley, M. I.; Quotients of Some Finite Universal Locally Projective Polytopes, Discrete and Computational Geometry 29 pp435-443 (2003)

to this polytope.