Polytope of Type {3,4,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,3}*1152
Also Known As : 24-cell, {3,4,3}. if this polytope has another name.
Group : SmallGroup(1152,157478)
Rank : 4
Schlafli Type : {3,4,3}
Number of vertices, edges, etc : 24, 96, 96, 24
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,4,3}*576
32-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1> of order 2.
14 facets:
4 of 2-fold non-regular quotient of {3,4}*48
10 of {3,4}*48
14 vertex figures:
10 of {4,3}*48
4 of 2-fold non-regular quotient of {4,3}*48
P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s2*s3*s2*s1*s0*s2*s1*s2*s3*s2*s1*s0*s2*s1*s2*s3> of order 2.
12 facets:
12 of {3,4}*48
13 vertex figures:
11 of {4,3}*48
2 of {4,3}*24
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
13 facets:
2 of {3,4}*24
11 of {3,4}*48
12 vertex figures:
12 of {4,3}*48
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s3*s2*s1*s0*s2*s1*s2*s3> of order 3.
8 facets:
8 of {3,4}*48
8 vertex figures:
8 of {4,3}*48
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1, s1*s2*s1*s2*s3*s2*s1*s0*s2*s1*s2*s3*s2*s1*s0*s2*s1*s2*s3> of order 4.
8 facets:
2 of 2-fold non-regular quotient of {3,4}*48
2 of {3,4}*24
4 of {3,4}*48
7 vertex figures:
5 of {4,3}*48
2 of 2-fold non-regular quotient of {4,3}*48
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 4.
9 facets:
2 of {3,2}*12
3 of 2-fold non-regular quotient of {3,4}*48
4 of {3,4}*48
9 vertex figures:
6 of 2-fold non-regular quotient of {4,3}*48
3 of {4,3}*48
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1, s0*s2*s3*s2*s1*s0*s2*s1*s2*s3*s2*s1*s0*s2*s1*s2*s3> of order 4.
7 facets:
2 of 2-fold non-regular quotient of {3,4}*48
5 of {3,4}*48
8 vertex figures:
4 of {4,3}*48
2 of 2-fold non-regular quotient of {4,3}*48
2 of {4,3}*24
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1, s0*s2*s1*s0*s3*s2*s1*s0*s2*s1*s3*s2> of order 4.
9 facets:
6 of 2-fold non-regular quotient of {3,4}*48
3 of {3,4}*48
9 vertex figures:
4 of {4,3}*48
3 of 2-fold non-regular quotient of {4,3}*48
2 of {2,3}*12
Permutation Representation (GAP) :
s0 := ( 5, 8)( 6, 7)(11,13)(12,14)(19,24)(20,23);;
s1 := ( 3, 8)( 4, 7)(13,16)(14,15)(19,21)(20,22);;
s2 := ( 3, 4)( 9,22)(10,21)(11,20)(12,19)(13,23)(14,24)(15,18)(16,17);;
s3 := ( 1,10)( 2, 9)( 3,16)( 4,15)( 5,11)( 6,12)( 7,14)( 8,13)(17,18);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(24)!( 5, 8)( 6, 7)(11,13)(12,14)(19,24)(20,23);
s1 := Sym(24)!( 3, 8)( 4, 7)(13,16)(14,15)(19,21)(20,22);
s2 := Sym(24)!( 3, 4)( 9,22)(10,21)(11,20)(12,19)(13,23)(14,24)(15,18)(16,17);
s3 := Sym(24)!( 1,10)( 2, 9)( 3,16)( 4,15)( 5,11)( 6,12)( 7,14)( 8,13)(17,18);
poly := sub<Sym(24)|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 >;
References : - Schläfli, L.; Theorie Der Vielfachen Kontinuität, Denkschriften Der Schweizerischen Naturforschenden Gesellschaft, 38, pp1–237 (1901)
to this polytope