Questions?
See the FAQ
or other info.

Polytope of Type {4,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,4}*96
Also Known As : {{4,3}3,{3,4}3}. if this polytope has another name.
Group : SmallGroup(96,227)
Rank : 4
Schlafli Type : {4,3,4}
Number of vertices, edges, etc : 4, 6, 6, 4
Order of s0s1s2s3 : 3
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Locally Projective
Non-Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,3,4,2} of size 192
{4,3,4,4} of size 768
Vertex Figure Of :
{2,4,3,4} of size 192
{4,4,3,4} of size 768
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,3,4}*192a, {4,3,4}*192b, {4,6,4}*192d, {4,6,4}*192e, {4,6,4}*192f, {4,6,4}*192g
3-fold covers : {4,9,4}*288
4-fold covers : {4,12,4}*384f, {4,12,4}*384g, {4,12,4}*384h, {4,12,4}*384i, {4,3,8}*384, {8,3,4}*384, {4,3,4}*384, {4,6,4}*384c, {4,6,4}*384d, {4,6,4}*384e, {4,6,4}*384f
5-fold covers : {4,15,4}*480
6-fold covers : {4,9,4}*576a, {4,9,4}*576b, {4,18,4}*576d, {4,18,4}*576e, {4,18,4}*576f, {4,18,4}*576g, {4,3,12}*576, {4,6,12}*576f, {12,3,4}*576, {12,6,4}*576f
7-fold covers : {4,21,4}*672
8-fold covers : {4,6,4}*768e, {4,6,4}*768f, {4,3,8}*768a, {4,6,8}*768d, {8,3,4}*768a, {8,6,4}*768d, {4,24,4}*768m, {4,24,4}*768n, {4,24,4}*768o, {4,24,4}*768p, {4,6,4}*768g, {4,6,4}*768h, {4,12,4}*768i, {4,12,4}*768j, {4,12,4}*768k, {4,12,4}*768l, {4,12,4}*768m, {4,12,4}*768n, {4,12,4}*768o, {4,12,4}*768p, {4,3,8}*768b, {4,6,8}*768e, {4,6,8}*768f, {8,3,4}*768b, {8,6,4}*768e, {8,6,4}*768f, {4,6,8}*768g, {8,6,4}*768g, {4,3,8}*768c, {4,6,8}*768h, {8,3,4}*768c, {8,6,4}*768h, {4,3,4}*768, {4,6,4}*768i, {4,6,4}*768j, {4,6,4}*768k, {4,6,4}*768l
9-fold covers : {4,27,4}*864
10-fold covers : {4,6,20}*960c, {20,6,4}*960c, {4,15,4}*960a, {4,15,4}*960b, {4,30,4}*960d, {4,30,4}*960e, {4,30,4}*960f, {4,30,4}*960g
11-fold covers : {4,33,4}*1056
12-fold covers : {4,36,4}*1152f, {4,36,4}*1152g, {4,36,4}*1152h, {4,36,4}*1152i, {4,9,8}*1152, {8,9,4}*1152, {4,9,4}*1152, {4,18,4}*1152c, {4,18,4}*1152d, {4,18,4}*1152e, {4,18,4}*1152f, {4,3,24}*1152, {24,3,4}*1152, {4,3,12}*1152b, {4,6,12}*1152e, {4,6,12}*1152f, {4,6,12}*1152g, {4,6,12}*1152h, {12,3,4}*1152b, {12,6,4}*1152e, {12,6,4}*1152f, {12,6,4}*1152g, {12,6,4}*1152h
13-fold covers : {4,39,4}*1248
14-fold covers : {4,6,28}*1344c, {28,6,4}*1344c, {4,21,4}*1344a, {4,21,4}*1344b, {4,42,4}*1344d, {4,42,4}*1344e, {4,42,4}*1344f, {4,42,4}*1344g
15-fold covers : {4,45,4}*1440
17-fold covers : {4,51,4}*1632
18-fold covers : {4,27,4}*1728a, {4,27,4}*1728b, {4,54,4}*1728d, {4,54,4}*1728e, {4,54,4}*1728f, {4,54,4}*1728g, {4,6,36}*1728d, {36,6,4}*1728d, {4,9,12}*1728, {4,18,12}*1728e, {12,9,4}*1728, {12,18,4}*1728e, {4,3,12}*1728, {4,6,12}*1728i, {12,3,4}*1728, {12,6,4}*1728i
19-fold covers : {4,57,4}*1824
20-fold covers : {4,60,4}*1920f, {4,60,4}*1920g, {4,60,4}*1920h, {4,60,4}*1920i, {4,15,8}*1920, {8,15,4}*1920, {4,6,20}*1920c, {4,6,20}*1920d, {20,6,4}*1920c, {20,6,4}*1920d, {4,15,4}*1920c, {4,30,4}*1920c, {4,30,4}*1920d, {4,30,4}*1920e, {4,30,4}*1920f
Permutation Representation (GAP) :
```s0 := ( 1, 2)( 3, 6)( 4, 5)( 7,14)( 8,15)( 9,10)(11,13)(12,16);;
s1 := ( 2, 4)( 3, 7)( 6,11)( 9,14)(10,13)(12,15);;
s2 := ( 3, 8)( 4, 5)( 6,15)( 9,16)(10,12)(11,13);;
s3 := ( 1, 8)( 2,15)( 3, 7)( 4,12)( 5,16)( 6,14)( 9,11)(10,13);;
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*s0*s1*s2*s0*s1*s2*s0*s1, s1*s3*s2*s1*s3*s2*s1*s3*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(16)!( 1, 2)( 3, 6)( 4, 5)( 7,14)( 8,15)( 9,10)(11,13)(12,16);
s1 := Sym(16)!( 2, 4)( 3, 7)( 6,11)( 9,14)(10,13)(12,15);
s2 := Sym(16)!( 3, 8)( 4, 5)( 6,15)( 9,16)(10,12)(11,13);
s3 := Sym(16)!( 1, 8)( 2,15)( 3, 7)( 4,12)( 5,16)( 6,14)( 9,11)(10,13);
poly := sub<Sym(16)|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*s0*s1*s2*s0*s1*s2*s0*s1,
s1*s3*s2*s1*s3*s2*s1*s3*s2 >;

```
References : None.
to this polytope