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# Polytope of Type {6,4,2}

Atlas Canonical Name : {6,4,2}*144
if this polytope has a name.
Group : SmallGroup(144,186)
Rank : 4
Schlafli Type : {6,4,2}
Number of vertices, edges, etc : 9, 18, 6, 2
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,4,2,2} of size 288
{6,4,2,3} of size 432
{6,4,2,4} of size 576
{6,4,2,5} of size 720
{6,4,2,6} of size 864
{6,4,2,7} of size 1008
{6,4,2,8} of size 1152
{6,4,2,9} of size 1296
{6,4,2,10} of size 1440
{6,4,2,11} of size 1584
{6,4,2,12} of size 1728
{6,4,2,13} of size 1872
Vertex Figure Of :
{2,6,4,2} of size 288
{4,6,4,2} of size 1440
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,4,4}*288, {6,4,2}*288
3-fold covers : {6,4,2}*432, {6,12,2}*432a, {6,12,2}*432b, {6,4,6}*432a, {6,12,2}*432c
4-fold covers : {6,4,8}*576, {6,8,2}*576, {6,4,4}*576, {12,4,2}*576
5-fold covers : {6,4,10}*720, {6,20,2}*720
6-fold covers : {6,4,4}*864a, {6,12,4}*864a, {6,12,4}*864b, {6,4,2}*864a, {6,12,2}*864e, {6,12,2}*864f, {6,4,12}*864, {6,12,4}*864c, {6,4,6}*864b, {6,4,2}*864b, {6,12,2}*864h, {6,12,2}*864i
7-fold covers : {6,4,14}*1008, {6,28,2}*1008
8-fold covers : {6,4,16}*1152, {12,4,4}*1152, {6,4,8}*1152a, {24,4,2}*1152a, {12,8,2}*1152a, {6,8,4}*1152a, {24,4,2}*1152b, {6,4,8}*1152b, {12,8,2}*1152b, {6,8,4}*1152b, {12,4,2}*1152, {6,4,4}*1152a, {6,16,2}*1152
9-fold covers : {18,4,2}*1296, {6,4,18}*1296, {6,4,6}*1296a, {6,12,6}*1296a, {6,12,6}*1296b, {6,36,2}*1296a, {6,12,2}*1296, {6,36,2}*1296b, {6,36,2}*1296c, {6,12,6}*1296e, {6,12,6}*1296g, {6,12,6}*1296h
10-fold covers : {6,4,20}*1440, {6,20,4}*1440, {6,4,10}*1440c, {30,4,2}*1440, {6,20,2}*1440
11-fold covers : {6,4,22}*1584, {6,44,2}*1584
12-fold covers : {6,4,8}*1728, {6,12,8}*1728a, {6,12,8}*1728b, {6,8,2}*1728a, {6,24,2}*1728d, {6,24,2}*1728e, {6,4,4}*1728a, {6,12,4}*1728h, {6,12,4}*1728i, {12,4,2}*1728b, {12,12,2}*1728f, {12,12,2}*1728g, {6,4,24}*1728, {6,12,8}*1728c, {6,8,6}*1728a, {6,4,12}*1728a, {12,4,2}*1728d, {12,12,2}*1728j, {6,8,2}*1728b, {6,24,2}*1728g, {6,4,4}*1728c, {6,12,4}*1728o, {12,4,6}*1728b, {6,24,2}*1728h, {6,12,4}*1728q, {12,12,2}*1728l, {6,12,6}*1728c, {6,12,4}*1728r, {12,12,2}*1728o
13-fold covers : {6,4,26}*1872, {6,52,2}*1872
Permutation Representation (GAP) :
```s0 := (3,4)(5,6);;
s1 := (2,3);;
s2 := (1,2)(3,5)(4,6);;
s3 := (7,8);;
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, s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(8)!(3,4)(5,6);
s1 := Sym(8)!(2,3);
s2 := Sym(8)!(1,2)(3,5)(4,6);
s3 := Sym(8)!(7,8);
poly := sub<Sym(8)|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,
s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

```

to this polytope