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# Polytope of Type {2,8,4}

Atlas Canonical Name : {2,8,4}*128b
if this polytope has a name.
Group : SmallGroup(128,1746)
Rank : 4
Schlafli Type : {2,8,4}
Number of vertices, edges, etc : 2, 8, 16, 4
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,8,4,2} of size 256
{2,8,4,4} of size 512
{2,8,4,6} of size 768
{2,8,4,10} of size 1280
{2,8,4,14} of size 1792
Vertex Figure Of :
{2,2,8,4} of size 256
{3,2,8,4} of size 384
{5,2,8,4} of size 640
{6,2,8,4} of size 768
{7,2,8,4} of size 896
{9,2,8,4} of size 1152
{10,2,8,4} of size 1280
{11,2,8,4} of size 1408
{13,2,8,4} of size 1664
{14,2,8,4} of size 1792
{15,2,8,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,4}*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 : {2,8,4}*256a, {2,8,8}*256a, {2,8,8}*256d, {4,8,4}*256b
3-fold covers : {2,24,4}*384b, {2,8,12}*384b, {6,8,4}*384b
4-fold covers : {2,8,8}*512a, {4,8,8}*512c, {4,8,8}*512d, {8,8,4}*512g, {8,8,4}*512h, {4,8,4}*512b, {4,8,4}*512c, {2,8,4}*512a, {2,8,8}*512d, {2,16,4}*512a, {2,16,4}*512b, {2,16,8}*512a, {2,16,8}*512b, {2,8,16}*512c, {2,8,16}*512e
5-fold covers : {2,40,4}*640b, {2,8,20}*640b, {10,8,4}*640b
6-fold covers : {6,8,4}*768a, {2,8,12}*768a, {2,24,4}*768a, {6,8,8}*768a, {2,24,8}*768a, {2,8,24}*768b, {6,8,8}*768d, {2,8,24}*768d, {2,24,8}*768d, {12,8,4}*768b, {4,24,4}*768b, {4,8,12}*768c
7-fold covers : {2,56,4}*896b, {2,8,28}*896b, {14,8,4}*896b
9-fold covers : {18,8,4}*1152b, {2,8,36}*1152b, {2,72,4}*1152b, {6,8,12}*1152b, {6,24,4}*1152d, {6,24,4}*1152e, {6,24,4}*1152f, {2,24,12}*1152d, {2,24,12}*1152e, {2,24,12}*1152f, {2,8,4}*1152b, {2,24,4}*1152b, {2,8,12}*1152b, {6,8,4}*1152b
10-fold covers : {10,8,4}*1280a, {2,8,20}*1280a, {2,40,4}*1280a, {10,8,8}*1280a, {2,40,8}*1280a, {2,8,40}*1280b, {10,8,8}*1280d, {2,8,40}*1280d, {2,40,8}*1280d, {20,8,4}*1280b, {4,40,4}*1280b, {4,8,20}*1280c
11-fold covers : {22,8,4}*1408b, {2,8,44}*1408b, {2,88,4}*1408b
13-fold covers : {26,8,4}*1664b, {2,8,52}*1664b, {2,104,4}*1664b
14-fold covers : {14,8,4}*1792a, {2,8,28}*1792a, {2,56,4}*1792a, {14,8,8}*1792a, {2,56,8}*1792a, {2,8,56}*1792b, {14,8,8}*1792d, {2,8,56}*1792d, {2,56,8}*1792d, {28,8,4}*1792b, {4,56,4}*1792b, {4,8,28}*1792c
15-fold covers : {30,8,4}*1920b, {2,8,60}*1920b, {2,120,4}*1920b, {10,8,12}*1920b, {6,8,20}*1920b, {10,24,4}*1920b, {6,40,4}*1920b, {2,40,12}*1920b, {2,24,20}*1920b
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 6, 8)( 7,10)( 9,12)(13,16)(15,17);;
s2 := ( 3, 4)( 5, 7)( 6, 9)( 8,11)(10,13)(12,15)(14,17)(16,18);;
s3 := ( 4, 6)( 5, 8)( 7,10)(11,14)(13,17)(15,16);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```

to this polytope