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# Polytope of Type {16,4}

Atlas Canonical Name : {16,4}*128b
if this polytope has a name.
Group : SmallGroup(128,922)
Rank : 3
Schlafli Type : {16,4}
Number of vertices, edges, etc : 16, 32, 4
Order of s0s1s2 : 16
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{16,4,2} of size 256
{16,4,4} of size 512
{16,4,6} of size 768
{16,4,10} of size 1280
{16,4,14} of size 1792
Vertex Figure Of :
{2,16,4} of size 256
{4,16,4} of size 512
{4,16,4} of size 512
{6,16,4} of size 768
{10,16,4} of size 1280
{14,16,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,4}*64a
4-fold quotients : {4,4}*32, {8,2}*32
8-fold quotients : {2,4}*16, {4,2}*16
16-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {16,4}*256a, {16,8}*256e, {16,8}*256f
3-fold covers : {48,4}*384b, {16,12}*384b
4-fold covers : {16,4}*512a, {16,8}*512a, {16,16}*512e, {16,16}*512f, {16,16}*512h, {16,16}*512j, {16,8}*512c, {32,4}*512a, {32,4}*512b
5-fold covers : {80,4}*640b, {16,20}*640b
6-fold covers : {16,12}*768a, {48,4}*768a, {16,24}*768e, {48,8}*768e, {48,8}*768f, {16,24}*768f
7-fold covers : {112,4}*896b, {16,28}*896b
9-fold covers : {16,36}*1152b, {144,4}*1152b, {48,12}*1152d, {48,12}*1152e, {48,12}*1152f, {16,4}*1152b, {48,4}*1152b, {16,12}*1152b
10-fold covers : {16,20}*1280a, {80,4}*1280a, {16,40}*1280e, {80,8}*1280e, {80,8}*1280f, {16,40}*1280f
11-fold covers : {16,44}*1408b, {176,4}*1408b
13-fold covers : {16,52}*1664b, {208,4}*1664b
14-fold covers : {16,28}*1792a, {112,4}*1792a, {16,56}*1792e, {112,8}*1792e, {112,8}*1792f, {16,56}*1792f
15-fold covers : {16,60}*1920b, {240,4}*1920b, {80,12}*1920b, {48,20}*1920b
Permutation Representation (GAP) :
```s0 := ( 1, 9)( 2,10)( 3,12)( 4,11)( 5,15)( 6,16)( 7,13)( 8,14);;
s1 := ( 3, 4)( 5, 7)( 6, 8)( 9,13)(10,14)(11,16)(12,15);;
s2 := ( 5, 6)( 7, 8)(13,14)(15,16);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope