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# Polytope of Type {2,4,4}

Atlas Canonical Name : {2,4,4}*144
if this polytope has a name.
Group : SmallGroup(144,186)
Rank : 4
Schlafli Type : {2,4,4}
Number of vertices, edges, etc : 2, 9, 18, 9
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,4,2} of size 288
{2,4,4,3} of size 1440
{2,4,4,4} of size 1440
Vertex Figure Of :
{2,2,4,4} of size 288
{3,2,4,4} of size 432
{4,2,4,4} of size 576
{5,2,4,4} of size 720
{6,2,4,4} of size 864
{7,2,4,4} of size 1008
{8,2,4,4} of size 1152
{9,2,4,4} of size 1296
{10,2,4,4} of size 1440
{11,2,4,4} of size 1584
{12,2,4,4} of size 1728
{13,2,4,4} of size 1872
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,4,4}*288
3-fold covers : {2,4,12}*432, {2,12,4}*432
4-fold covers : {4,4,4}*576b, {2,4,4}*576
6-fold covers : {2,4,12}*864a, {2,12,4}*864a, {6,4,4}*864b, {2,4,12}*864b, {2,12,4}*864b
8-fold covers : {4,4,4}*1152a, {2,4,8}*1152a, {2,8,4}*1152a, {2,4,8}*1152b, {2,8,4}*1152b, {2,4,4}*1152, {8,4,4}*1152
9-fold covers : {2,4,4}*1296, {2,12,12}*1296, {6,4,4}*1296
10-fold covers : {2,4,20}*1440, {10,4,4}*1440, {2,20,4}*1440
12-fold covers : {4,4,12}*1728a, {4,12,4}*1728a, {2,4,12}*1728a, {2,12,4}*1728a, {12,4,4}*1728b, {2,4,12}*1728d, {2,12,4}*1728c, {6,4,4}*1728b, {4,12,4}*1728d, {4,4,12}*1728c, {2,12,12}*1728m
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (7,8);;
s2 := (3,4)(5,7)(6,8);;
s3 := (4,5);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
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);
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,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;

```

to this polytope