Polytope of Type {3,4,3} with group SmallGroup(576,8654)

Overview

Group: SmallGroup(576,8654)
Rank: 4
Schläfli Type: {3,4,3}
Vertices, edges, …: 12, 48, 48, 12
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 4
Also known as: hemi-24-cell, {3,4,3}₆. if this polytope has another name.

Special Properties

Projective
Locally Spherical
Orientable
Self-Dual

Quotients maximal quotients in bold

16-fold

{3,2,3}*36

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

8 facets

4 of 2-fold non-regular quotient of {3,4}*48
4 of {3,4}*48

8 vertex figures

4 of {4,3}*48
4 of 2-fold non-regular quotient of {4,3}*48

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

6 facets

4 of {3,2}*12
2 of {3,4}*48

6 vertex figures

6 of 2-fold non-regular quotient of {4,3}*48

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

6 facets

6 of 2-fold non-regular quotient of {3,4}*48

6 vertex figures

2 of {4,3}*48
4 of {2,3}*12

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

4 facets

2 of {3,2}*12
2 of 2-fold non-regular quotient of {3,4}*48

4 vertex figures

2 of 2-fold non-regular quotient of {4,3}*48
2 of {2,3}*12

Representations

Permutation Representation (GAP)

s0 := ( 2, 3)( 7, 8)(10,12);;
s1 := ( 2, 4)( 6, 7)(11,12);;
s2 := ( 5,11)( 6, 9)( 7,10)( 8,12);;
s3 := ( 1, 9)( 2,12)( 3,10)( 4,11);;
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, 
s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s3*s2*s3*s1*s2*s0*s1*s3*s2*s3*s1*s2*s0*s1*s2*s3*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(12)!( 2, 3)( 7, 8)(10,12);
s1 := Sym(12)!( 2, 4)( 6, 7)(11,12);
s2 := Sym(12)!( 5,11)( 6, 9)( 7,10)( 8,12);
s3 := Sym(12)!( 1, 9)( 2,12)( 3,10)( 4,11);
poly := sub<Sym(12)|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, s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s3*s2*s3*s1*s2*s0*s1*s3*s2*s3*s1*s2*s0*s1*s2*s3*s0*s1 >;

References

None.

to this polytope.