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# Polytope of Type {4,8,2,2}

Atlas Canonical Name : {4,8,2,2}*256b
if this polytope has a name.
Group : SmallGroup(256,53366)
Rank : 5
Schlafli Type : {4,8,2,2}
Number of vertices, edges, etc : 4, 16, 8, 2, 2
Order of s0s1s2s3s4 : 8
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,8,2,2,2} of size 512
{4,8,2,2,3} of size 768
{4,8,2,2,5} of size 1280
{4,8,2,2,7} of size 1792
Vertex Figure Of :
{2,4,8,2,2} of size 512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,2,2}*128
4-fold quotients : {2,4,2,2}*64, {4,2,2,2}*64
8-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,8,2,2}*512a, {8,8,2,2}*512c, {8,8,2,2}*512d, {4,8,4,2}*512c
3-fold covers : {4,8,2,6}*768b, {4,8,6,2}*768b, {12,8,2,2}*768b, {4,24,2,2}*768b
5-fold covers : {4,8,2,10}*1280b, {4,8,10,2}*1280b, {20,8,2,2}*1280b, {4,40,2,2}*1280b
7-fold covers : {4,8,2,14}*1792b, {4,8,14,2}*1792b, {28,8,2,2}*1792b, {4,56,2,2}*1792b
Permutation Representation (GAP) :
```s0 := ( 2, 4)( 3, 6)( 5, 8)( 9,12)(11,15)(13,14);;
s1 := ( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,11)(10,13)(12,15)(14,16);;
s2 := ( 2, 3)( 4, 6)( 5, 8)( 7,10)(11,14)(13,15);;
s3 := (17,18);;
s4 := (19,20);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(20)!( 2, 4)( 3, 6)( 5, 8)( 9,12)(11,15)(13,14);
s1 := Sym(20)!( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,11)(10,13)(12,15)(14,16);
s2 := Sym(20)!( 2, 3)( 4, 6)( 5, 8)( 7,10)(11,14)(13,15);
s3 := Sym(20)!(17,18);
s4 := Sym(20)!(19,20);
poly := sub<Sym(20)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4,
s0*s1*s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

```

to this polytope