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# Polytope of Type {4,16}

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

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

```
References : None.
to this polytope