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# Polytope of Type {4,4}

Atlas Canonical Name : {4,4}*128
Also Known As : {4,4}(4,0), {4,4|4}. if this polytope has another name.
Group : SmallGroup(128,928)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 16, 32, 16
Order of s0s1s2 : 8
Order of s0s1s2s1 : 4
Special Properties :
Toroidal
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Skewing Operation
Facet Of :
{4,4,2} of size 256
{4,4,3} of size 384
{4,4,4} of size 512
{4,4,6} of size 768
{4,4,3} of size 768
{4,4,6} of size 768
{4,4,6} of size 768
{4,4,3} of size 768
{4,4,9} of size 1152
{4,4,4} of size 1152
{4,4,6} of size 1152
{4,4,10} of size 1280
{4,4,14} of size 1792
{4,4,15} of size 1920
Vertex Figure Of :
{2,4,4} of size 256
{3,4,4} of size 384
{4,4,4} of size 512
{6,4,4} of size 768
{3,4,4} of size 768
{6,4,4} of size 768
{6,4,4} of size 768
{3,4,4} of size 768
{9,4,4} of size 1152
{4,4,4} of size 1152
{6,4,4} of size 1152
{10,4,4} of size 1280
{14,4,4} of size 1792
{15,4,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4}*64
4-fold quotients : {4,4}*32
8-fold quotients : {2,4}*16, {4,2}*16
16-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,8}*256a, {8,4}*256a, {4,4}*256, {4,8}*256c, {8,4}*256c
3-fold covers : {4,12}*384a, {12,4}*384a
4-fold covers : {4,16}*512a, {16,4}*512a, {8,8}*512c, {4,8}*512a, {8,4}*512a, {8,8}*512e, {8,8}*512f, {4,16}*512b, {16,4}*512b, {4,8}*512b, {4,8}*512c, {8,4}*512b, {8,4}*512c, {8,8}*512j, {8,8}*512k, {8,8}*512l, {8,8}*512n, {4,16}*512c, {4,16}*512d, {16,4}*512c, {16,4}*512d, {8,8}*512t, {4,4}*512, {4,8}*512d, {8,4}*512d
5-fold covers : {4,20}*640a, {20,4}*640a
6-fold covers : {8,12}*768a, {12,8}*768a, {4,24}*768a, {24,4}*768a, {4,12}*768a, {12,4}*768a, {8,12}*768c, {12,8}*768c, {4,24}*768c, {24,4}*768c
7-fold covers : {4,28}*896, {28,4}*896
9-fold covers : {4,36}*1152a, {36,4}*1152a, {12,12}*1152a, {12,12}*1152b, {12,12}*1152c, {4,12}*1152a, {12,4}*1152a, {4,4}*1152
10-fold covers : {8,20}*1280a, {20,8}*1280a, {4,40}*1280a, {40,4}*1280a, {4,20}*1280a, {20,4}*1280a, {8,20}*1280c, {20,8}*1280c, {4,40}*1280c, {40,4}*1280c
11-fold covers : {4,44}*1408, {44,4}*1408
13-fold covers : {4,52}*1664, {52,4}*1664
14-fold covers : {8,28}*1792a, {28,8}*1792a, {4,56}*1792a, {56,4}*1792a, {4,28}*1792, {28,4}*1792, {8,28}*1792c, {28,8}*1792c, {4,56}*1792c, {56,4}*1792c
15-fold covers : {4,60}*1920a, {60,4}*1920a, {12,20}*1920a, {20,12}*1920a
Permutation Representation (GAP) :
```s0 := ( 5, 7)( 6, 8)(13,15)(14,16);;
s1 := ( 3, 4)( 7, 8)( 9,13)(10,14)(11,16)(12,15);;
s2 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

```
References : None.
to this polytope