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# Polytope of Type {4,5}

Atlas Canonical Name : {4,5}*160
Also Known As : {4,5}5if this polytope has another name.
Group : SmallGroup(160,234)
Rank : 3
Schlafli Type : {4,5}
Number of vertices, edges, etc : 16, 40, 20
Order of s0s1s2 : 5
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,5,2} of size 320
{4,5,3} of size 960
{4,5,10} of size 1600
{4,5,4} of size 1920
{4,5,4} of size 1920
{4,5,6} of size 1920
{4,5,3} of size 1920
{4,5,6} of size 1920
{4,5,6} of size 1920
Vertex Figure Of :
{2,4,5} of size 320
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,5}*320a, {8,5}*320b, {4,5}*320, {4,10}*320a, {4,10}*320b
3-fold covers : {4,15}*480
4-fold covers : {4,20}*640b, {4,20}*640c, {8,5}*640a, {8,10}*640a, {8,10}*640b, {8,10}*640c, {8,10}*640d, {4,5}*640, {4,10}*640a, {4,20}*640d, {4,20}*640e, {8,5}*640b, {4,10}*640b
5-fold covers : {4,25}*800
6-fold covers : {8,15}*960b, {8,15}*960c, {12,10}*960e, {4,15}*960, {4,30}*960c, {4,30}*960d
7-fold covers : {4,35}*1120
8-fold covers : {8,5}*1280, {8,10}*1280a, {8,10}*1280b, {8,20}*1280e, {8,20}*1280f, {8,20}*1280g, {8,20}*1280h, {8,20}*1280i, {8,20}*1280j, {8,20}*1280k, {8,20}*1280l, {4,40}*1280e, {4,40}*1280f, {4,40}*1280g, {4,40}*1280h, {4,10}*1280a, {4,20}*1280b, {4,20}*1280c, {8,10}*1280c, {4,10}*1280b, {4,20}*1280d, {8,10}*1280d, {4,20}*1280e, {4,10}*1280c, {8,10}*1280e, {8,10}*1280f
9-fold covers : {4,45}*1440
10-fold covers : {8,25}*1600a, {8,25}*1600b, {4,25}*1600, {4,50}*1600a, {4,50}*1600b, {20,5}*1600, {20,10}*1600
11-fold covers : {4,55}*1760
12-fold covers : {12,20}*1920d, {12,20}*1920e, {24,10}*1920a, {24,10}*1920b, {4,60}*1920f, {4,60}*1920g, {8,15}*1920b, {8,30}*1920h, {8,30}*1920i, {8,30}*1920j, {8,30}*1920k, {4,15}*1920a, {4,30}*1920c, {4,60}*1920h, {4,60}*1920i, {8,15}*1920c, {4,15}*1920b, {12,10}*1920a, {4,30}*1920d
Permutation Representation (GAP) :
```s0 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16);;
s1 := ( 2, 9)( 3,12)( 5,15)( 6, 7)( 8,14)(13,16);;
s2 := ( 3, 4)( 5, 6)( 9,16)(10,15)(11,13)(12,14);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope