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# Polytope of Type {4,8}

Atlas Canonical Name : {4,8}*128a
if this polytope has a name.
Group : SmallGroup(128,327)
Rank : 3
Schlafli Type : {4,8}
Number of vertices, edges, etc : 8, 32, 16
Order of s0s1s2 : 8
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,8,2} of size 256
{4,8,4} of size 512
{4,8,4} of size 512
{4,8,6} of size 768
{4,8,3} of size 768
{4,8,3} of size 768
{4,8,10} of size 1280
{4,8,14} of size 1792
Vertex Figure Of :
{2,4,8} of size 256
{4,4,8} of size 512
{6,4,8} of size 768
{10,4,8} of size 1280
{14,4,8} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64
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 : {8,8}*256a, {4,8}*256a, {8,8}*256c, {4,16}*256a, {4,16}*256b
3-fold covers : {4,24}*384a, {12,8}*384a
4-fold covers : {4,16}*512a, {8,16}*512a, {8,16}*512b, {8,16}*512c, {16,8}*512c, {8,16}*512d, {16,8}*512d, {8,16}*512e, {16,8}*512e, {8,16}*512f, {16,8}*512f, {8,8}*512a, {8,8}*512b, {8,8}*512c, {4,8}*512a, {8,8}*512e, {4,16}*512b, {4,8}*512b, {4,8}*512c, {8,8}*512j, {8,8}*512k, {4,16}*512c, {4,16}*512d, {8,8}*512p, {8,8}*512r, {8,16}*512g, {8,16}*512h, {4,32}*512a, {4,32}*512b
5-fold covers : {4,40}*640a, {20,8}*640a
6-fold covers : {8,24}*768a, {24,8}*768a, {12,8}*768a, {4,24}*768a, {24,8}*768c, {8,24}*768d, {12,16}*768a, {4,48}*768a, {12,16}*768b, {4,48}*768b
7-fold covers : {4,56}*896a, {28,8}*896a
9-fold covers : {36,8}*1152a, {4,72}*1152a, {12,24}*1152a, {12,24}*1152b, {12,24}*1152c, {4,8}*1152a, {4,24}*1152a, {12,8}*1152a
10-fold covers : {8,40}*1280a, {40,8}*1280a, {20,8}*1280a, {4,40}*1280a, {40,8}*1280c, {8,40}*1280d, {20,16}*1280a, {4,80}*1280a, {20,16}*1280b, {4,80}*1280b
11-fold covers : {44,8}*1408a, {4,88}*1408a
13-fold covers : {52,8}*1664a, {4,104}*1664a
14-fold covers : {8,56}*1792a, {56,8}*1792a, {28,8}*1792a, {4,56}*1792a, {56,8}*1792c, {8,56}*1792d, {28,16}*1792a, {4,112}*1792a, {28,16}*1792b, {4,112}*1792b
15-fold covers : {60,8}*1920a, {4,120}*1920a, {12,40}*1920a, {20,24}*1920a
Permutation Representation (GAP) :
```s0 := ( 5, 6)( 7, 8)(13,14)(15,16);;
s1 := ( 5, 7)( 6, 8)( 9,13)(10,14)(11,15)(12,16);;
s2 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope