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# Polytope of Type {2,4,8}

Atlas Canonical Name : {2,4,8}*128a
if this polytope has a name.
Group : SmallGroup(128,1728)
Rank : 4
Schlafli Type : {2,4,8}
Number of vertices, edges, etc : 2, 4, 16, 8
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,8,2} of size 256
{2,4,8,4} of size 512
{2,4,8,4} of size 512
{2,4,8,6} of size 768
{2,4,8,3} of size 768
{2,4,8,10} of size 1280
{2,4,8,14} of size 1792
Vertex Figure Of :
{2,2,4,8} of size 256
{3,2,4,8} of size 384
{5,2,4,8} of size 640
{6,2,4,8} of size 768
{7,2,4,8} of size 896
{9,2,4,8} of size 1152
{10,2,4,8} of size 1280
{11,2,4,8} of size 1408
{13,2,4,8} of size 1664
{14,2,4,8} of size 1792
{15,2,4,8} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,4}*64, {2,2,8}*64
4-fold quotients : {2,2,4}*32, {2,4,2}*32
8-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,4,8}*256a, {2,8,8}*256a, {2,8,8}*256b, {4,4,8}*256a, {2,4,16}*256a, {2,4,16}*256b
3-fold covers : {2,4,24}*384a, {2,12,8}*384a, {6,4,8}*384a
4-fold covers : {2,8,8}*512a, {8,4,8}*512b, {4,8,8}*512a, {4,4,8}*512a, {4,8,8}*512c, {4,8,8}*512e, {4,8,8}*512g, {4,4,8}*512b, {8,4,8}*512d, {2,4,8}*512a, {2,8,8}*512c, {2,4,16}*512a, {2,4,16}*512b, {2,16,8}*512a, {2,16,8}*512b, {2,8,16}*512c, {2,8,16}*512d, {2,16,8}*512d, {2,8,16}*512e, {2,8,16}*512f, {2,16,8}*512f, {4,4,16}*512a, {4,4,16}*512b, {2,4,32}*512a, {2,4,32}*512b
5-fold covers : {2,4,40}*640a, {2,20,8}*640a, {10,4,8}*640a
6-fold covers : {6,4,8}*768a, {2,12,8}*768a, {2,4,24}*768a, {6,8,8}*768a, {6,8,8}*768b, {2,24,8}*768a, {2,8,24}*768b, {2,8,24}*768c, {2,24,8}*768c, {12,4,8}*768a, {4,12,8}*768a, {4,4,24}*768a, {6,4,16}*768a, {2,12,16}*768a, {2,4,48}*768a, {6,4,16}*768b, {2,12,16}*768b, {2,4,48}*768b
7-fold covers : {2,4,56}*896a, {2,28,8}*896a, {14,4,8}*896a
9-fold covers : {18,4,8}*1152a, {2,36,8}*1152a, {2,4,72}*1152a, {6,12,8}*1152a, {6,12,8}*1152b, {6,12,8}*1152c, {6,4,24}*1152a, {2,12,24}*1152a, {2,12,24}*1152b, {2,12,24}*1152c, {6,4,8}*1152a, {2,4,8}*1152a, {2,4,24}*1152a, {2,12,8}*1152a
10-fold covers : {10,4,8}*1280a, {2,20,8}*1280a, {2,4,40}*1280a, {10,8,8}*1280a, {10,8,8}*1280b, {2,40,8}*1280a, {2,8,40}*1280b, {2,8,40}*1280c, {2,40,8}*1280c, {20,4,8}*1280a, {4,20,8}*1280a, {4,4,40}*1280a, {10,4,16}*1280a, {2,20,16}*1280a, {2,4,80}*1280a, {10,4,16}*1280b, {2,20,16}*1280b, {2,4,80}*1280b
11-fold covers : {22,4,8}*1408a, {2,44,8}*1408a, {2,4,88}*1408a
13-fold covers : {26,4,8}*1664a, {2,52,8}*1664a, {2,4,104}*1664a
14-fold covers : {14,4,8}*1792a, {2,28,8}*1792a, {2,4,56}*1792a, {14,8,8}*1792a, {14,8,8}*1792b, {2,56,8}*1792a, {2,8,56}*1792b, {2,8,56}*1792c, {2,56,8}*1792c, {28,4,8}*1792a, {4,28,8}*1792a, {4,4,56}*1792a, {14,4,16}*1792a, {2,28,16}*1792a, {2,4,112}*1792a, {14,4,16}*1792b, {2,28,16}*1792b, {2,4,112}*1792b
15-fold covers : {30,4,8}*1920a, {2,60,8}*1920a, {2,4,120}*1920a, {10,12,8}*1920a, {6,20,8}*1920a, {10,4,24}*1920a, {6,4,40}*1920a, {2,12,40}*1920a, {2,20,24}*1920a
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 6)( 5, 8)(12,15)(14,17);;
s2 := ( 3, 4)( 5, 7)( 6, 9)( 8,11)(10,12)(13,15)(14,16)(17,18);;
s3 := ( 4, 5)( 6, 8)( 7,10)(11,13)(12,14)(15,17);;
poly := Group([s0,s1,s2,s3]);;

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

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

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

```

to this polytope