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# Polytope of Type {6,3,4}

Atlas Canonical Name : {6,3,4}*144
if this polytope has a name.
Group : SmallGroup(144,183)
Rank : 4
Schlafli Type : {6,3,4}
Number of vertices, edges, etc : 6, 9, 6, 4
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,3,4,2} of size 288
{6,3,4,4} of size 1152
Vertex Figure Of :
{2,6,3,4} of size 288
{3,6,3,4} of size 432
{4,6,3,4} of size 576
{6,6,3,4} of size 864
{6,6,3,4} of size 864
{8,6,3,4} of size 1152
{9,6,3,4} of size 1296
{3,6,3,4} of size 1296
{10,6,3,4} of size 1440
{12,6,3,4} of size 1728
{12,6,3,4} of size 1728
{4,6,3,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,3,4}*288, {6,6,4}*288e, {6,6,4}*288f
3-fold covers : {6,9,4}*432, {6,3,4}*432
4-fold covers : {6,12,4}*576f, {6,12,4}*576g, {6,3,8}*576, {12,6,4}*576e, {6,3,4}*576, {6,6,4}*576b, {12,3,4}*576
5-fold covers : {6,15,4}*720
6-fold covers : {6,9,4}*864, {6,18,4}*864d, {6,18,4}*864e, {6,3,4}*864, {6,6,4}*864e, {6,6,4}*864f, {6,3,12}*864, {6,6,4}*864i, {6,6,12}*864h
7-fold covers : {6,21,4}*1008
8-fold covers : {6,6,4}*1152b, {6,3,8}*1152, {6,6,8}*1152a, {6,24,4}*1152i, {6,24,4}*1152j, {12,12,4}*1152f, {12,12,4}*1152g, {24,6,4}*1152e, {12,3,4}*1152a, {6,12,4}*1152f, {6,6,4}*1152d, {6,12,4}*1152h, {6,6,8}*1152c, {6,6,8}*1152e, {12,6,4}*1152d, {24,3,4}*1152, {6,3,4}*1152b, {6,6,4}*1152g, {6,6,4}*1152h, {12,3,4}*1152b, {12,6,4}*1152g, {12,6,4}*1152h
9-fold covers : {6,27,4}*1296, {18,9,4}*1296, {18,3,4}*1296, {6,3,4}*1296a, {6,9,4}*1296a, {6,9,4}*1296b, {6,9,4}*1296c, {6,9,4}*1296d
10-fold covers : {6,6,20}*1440d, {30,6,4}*1440d, {6,15,4}*1440b, {6,30,4}*1440e, {6,30,4}*1440f
11-fold covers : {6,33,4}*1584
12-fold covers : {6,36,4}*1728e, {6,36,4}*1728f, {6,12,4}*1728d, {6,12,4}*1728e, {6,9,8}*1728, {6,3,8}*1728, {12,18,4}*1728d, {12,6,4}*1728e, {6,9,4}*1728, {6,3,4}*1728, {6,18,4}*1728b, {6,6,4}*1728a, {12,9,4}*1728, {12,3,4}*1728, {6,3,24}*1728, {6,12,4}*1728l, {6,12,4}*1728m, {12,6,4}*1728j, {6,6,4}*1728c, {6,6,12}*1728b, {6,6,12}*1728d
13-fold covers : {6,39,4}*1872
Permutation Representation (GAP) :
```s0 := ( 5, 9)( 6,10)( 7,11)( 8,12);;
s1 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12);;
s2 := ( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s3*s2*s1*s3*s2*s1*s3*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(12)!( 5, 9)( 6,10)( 7,11)( 8,12);
s1 := Sym(12)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)(11,12);
s2 := Sym(12)!( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12);
s3 := Sym(12)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12);
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,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s3*s2*s1*s3*s2*s1*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

```
References : None.
to this polytope