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# Polytope of Type {8,4}

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

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

```
References : None.
to this polytope