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# Polytope of Type {2,4,6}

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

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

```

to this polytope