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# Polytope of Type {4,8,2}

Atlas Canonical Name : {4,8,2}*128b
if this polytope has a name.
Group : SmallGroup(128,1746)
Rank : 4
Schlafli Type : {4,8,2}
Number of vertices, edges, etc : 4, 16, 8, 2
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,8,2,2} of size 256
{4,8,2,3} of size 384
{4,8,2,5} of size 640
{4,8,2,6} of size 768
{4,8,2,7} of size 896
{4,8,2,9} of size 1152
{4,8,2,10} of size 1280
{4,8,2,11} of size 1408
{4,8,2,13} of size 1664
{4,8,2,14} of size 1792
{4,8,2,15} of size 1920
Vertex Figure Of :
{2,4,8,2} of size 256
{4,4,8,2} of size 512
{6,4,8,2} of size 768
{10,4,8,2} of size 1280
{14,4,8,2} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,2}*64
4-fold quotients : {2,4,2}*32, {4,2,2}*32
8-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,8,2}*256a, {8,8,2}*256c, {8,8,2}*256d, {4,8,4}*256c
3-fold covers : {4,24,2}*384b, {12,8,2}*384b, {4,8,6}*384b
4-fold covers : {8,8,2}*512a, {8,8,4}*512c, {8,8,4}*512d, {4,8,8}*512g, {4,8,8}*512h, {4,8,4}*512a, {4,8,4}*512d, {4,8,2}*512a, {8,8,2}*512c, {4,16,2}*512a, {4,16,2}*512b, {8,16,2}*512a, {8,16,2}*512b, {16,8,2}*512c, {16,8,2}*512e
5-fold covers : {4,40,2}*640b, {20,8,2}*640b, {4,8,10}*640b
6-fold covers : {4,8,6}*768a, {12,8,2}*768a, {4,24,2}*768a, {8,8,6}*768c, {8,24,2}*768a, {24,8,2}*768b, {8,8,6}*768d, {8,24,2}*768d, {24,8,2}*768d, {4,8,12}*768b, {4,24,4}*768c, {12,8,4}*768c
7-fold covers : {4,56,2}*896b, {28,8,2}*896b, {4,8,14}*896b
9-fold covers : {4,8,18}*1152b, {36,8,2}*1152b, {4,72,2}*1152b, {12,8,6}*1152b, {4,24,6}*1152d, {4,24,6}*1152e, {4,24,6}*1152f, {12,24,2}*1152d, {12,24,2}*1152e, {12,24,2}*1152f, {4,8,2}*1152b, {4,24,2}*1152b, {12,8,2}*1152b, {4,8,6}*1152b
10-fold covers : {4,8,10}*1280a, {20,8,2}*1280a, {4,40,2}*1280a, {8,8,10}*1280c, {8,40,2}*1280a, {40,8,2}*1280b, {8,8,10}*1280d, {8,40,2}*1280d, {40,8,2}*1280d, {4,8,20}*1280b, {4,40,4}*1280c, {20,8,4}*1280c
11-fold covers : {4,8,22}*1408b, {44,8,2}*1408b, {4,88,2}*1408b
13-fold covers : {4,8,26}*1664b, {52,8,2}*1664b, {4,104,2}*1664b
14-fold covers : {4,8,14}*1792a, {28,8,2}*1792a, {4,56,2}*1792a, {8,8,14}*1792c, {8,56,2}*1792a, {56,8,2}*1792b, {8,8,14}*1792d, {8,56,2}*1792d, {56,8,2}*1792d, {4,8,28}*1792b, {4,56,4}*1792c, {28,8,4}*1792c
15-fold covers : {4,8,30}*1920b, {60,8,2}*1920b, {4,120,2}*1920b, {12,8,10}*1920b, {20,8,6}*1920b, {4,24,10}*1920b, {4,40,6}*1920b, {12,40,2}*1920b, {20,24,2}*1920b
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);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
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(18)!( 2, 4)( 3, 6)( 5, 8)( 9,12)(11,15)(13,14);
s1 := Sym(18)!( 1, 2)( 3, 5)( 4, 7)( 6, 9)( 8,11)(10,13)(12,15)(14,16);
s2 := Sym(18)!( 2, 3)( 4, 6)( 5, 8)( 7,10)(11,14)(13,15);
s3 := Sym(18)!(17,18);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

```

to this polytope